~ '.
[J
" .. "
CRITICAL DEPOSIT VELOCITIESFOR
LOW-CONCENTRATION SOLID-LIQUID MIXTURES
by
Millard P. Robinson, Jr.
A Thesis
Presented to the Graduate Faculty
. of Lehigh University
in Candidacy for the Degree of
Master of Science
FRITZ ENGINEERINGl:ABORATORY LIBRARY..
Lehigh University
1971
ACKNOWLEDGEMENTS
My sincerest thanks are extended· to Dr. Walter H. Graf,
Dit"ector of the Hydraulics Division, Fritz Engineering Laboratory,
and advisor to my Master's Degree program, for his influential advice
and guidance throughout the research program. I would also like to
give mention of and special thanks to Mr. Oner Yuce1 for his unsullied
partnership throughout the research study.
The research program was partially sponsored by the Federal
Water Quality Office of the U.S.D.I. [grant number WP-01478 (11020 EKD)]
and by the Lehigh's Office of Research.
Thanks are due to Mr. Elias Dittbrenner for installation and
maintenance of the t~sting system, to Mrs. Jane Lenner for. typing the
entire manuscript, and to Mr. John Gera and Mrs. Sharon Balogh for the
drafting. ~
Dr. Lynn S. Beedle is the Director of the Fritz Engineering
Laboratory and Dr. David A. VanHorn is the Chairman of the Civil Engi
neering Department.
-iii-
TABLE OF CONTENTS
CERTIFICATE OF APPROVAL
ACKNOWLEDGEMENTS
TABLE OF' CONTENTS
LIST OF SYMBOLS
LIST OF FIGURES
LIST OF TABLES
ABSTRACT
1. INTRODUCTION TO THE PROBLEM
2. SOLIDS TRANSPORT IN PIPES
2.1 General Remarks on Solid-Liquid Mixture Flow2.2 The Critical Deposit Velocity, ''V
C''
2~2.l Definition and Significance2.2.2 Previous Investigations2.2.3 A Modified Froude Number Analysis
3. LEHIGH EXPERIMENTS
3.1 Facilities3.2 Measuring Techniques
ii
iii
iv
vi
viii
x
1
2
4
411
111421
25
2534
3.2.13.2.2
Clear-Water TestsThe Loop System
3435
3.3 Description of Experiments
3.3.1 Range of Parameters Tested3.3.2 Testing Procedure
4. EVALUATION OF EXPERIMENTAL DATA
4.1 Analysis of Lehigh Results4.2 Comparison to Other· Data4.3 Engineering Application
39
3943
46
465764
4.3.1 Economics of Solid-Liquid Transport Systems 654.3.2 Application of the Lehigh Findings to Design 67
-iv-
/,I
/
5. CONCLUS IONS
TABLE OF CONTENTS (Continued)
71
.APPENDIX A - Evaluation of Loop Readings from ProgrammedOutput 74
APPENDIX B - Test Data Compilation 81
(
APPENDIX C Correlation Data
REFERENCES
VITA
\ "
".. ..
-v-
148
157
161
LIST OF SYMBOLS
/a correlation exponent, coefficient
A cross-sectional area of the pipe
b correlation exponent, coefficient
C moving volumetric solids concentration
c correlation exponent, coefficient
CR
solids concentration in the "riser" pipe
CD solids concentration in the "downcomer" pipe
D diameter of pipe (I.D.)
'd effective diameter of the sediment particles
d correlation exponent
~o mean diameter of the sediment particles
des solid's particle diameter (Sinclair)
dgo/dso non-uniformity coefficient of grain distribution
f friction factor
fm
mixture flow friction factor
I~ liquid flow friction factor
f,f1,f
2,f
3functions
FL
modified Froude number (Durand)
f function of correlation (Sinclair)s
head loss of the total mixture
head loss due to liquid component only
head loss due to solids component only
coefficient (Durand)
g
if1h
or -m M-mi~
i s
~
Fr(I),···Fr(IV) tested modified Froude numbers
gravitational acceleration
-vi..
,
I K
/~' ka,Is
" "ka ,k4,k
4,
ka,ks
'L
~
Qs
Q's
~
'IRe
s s
t'
V
Vc' "
Vm
vss
Vmax
epD
p
v
e
tan e
Ps
ys
~
~
l'e
correlation parameter (Wilson)
correlation coefficients
head loss length in both riser and downcomer pipes
mixture flowrate
volumetric solids delivery
optimum solids throughput
hydraulic radius
Reynolds number of the mean flow
specific gravity of solids
temperature
average velocity
critical deposit velocity
mixture velocity
settling velocity of sediment
maximum limiting deposit velocity (Sinclair)
dimensionless transport parameter (Durand)
carrying fluid density
kinematic fluid viscosity
pipe roughness
pipe slope
solids particle density
particle shapes; sphericity
head loss in the riser section
head loss in the downcomer section
pressure gradient for mixture f10wrate (Einstein)
-vii-
...
Figure
2.1
2.2
2.3
2.4
2.5
2.6
2.7
3.1
3.2
3.3
3.4
3.5
3.6 .
4.1
LIST OF FIGURES
Title
Regimes of Flow
Equi-Concentration Lines
Bed Motion of Plastic Pellets in a 6-Inch Pipe at theCritical Deposit Velocity
The Modified Froude Number, FL' versus Solids Concentration and Particle Diameter
Modified Froude Number versus Concentration; ParticleDiameter as Parameter
Modified Froude Number versus Particle Diameter; Concentration as Parameter
plot of Eq. (2.13); the Modified Froude Number Re lationship
Solid-Liquid Transport Test System
Setup for Tests in a Horizontal 4-Inch DiameterGalvanized Pipe
Low Flow Dune Transport of Coarse Sand Particles inthe Deposit Regime
Sediment Feed and Removal Facility
Loop System Charts
Photographic Representation of the Three Types ofSolid Particles Investigated, (a) Coarse Sand Particles, (b) Fine Sand Particles, and (c) PlasticPellets
Experimental Data from Lehigh Sand-Water and PlasticPellet-Water Studies; Modified Froude Number versusConcentration, Particle Diameter as Parameter
6
10
12
16
.18
18
23
26
29
29
31
37
40
50
•4.2 Best-Fit Equations for Lehigh's Sand-Water Data
Only; Modified Froude Number versus Concentration,Particle Diameter as Parameter
-viii-
54
}
Figure
4.3
4.4
4.5
LIST OF FIGURES (Continued)
Title
Modified Froude Number versus Solids Concentration,Particle Diameter as Parameter (Data from Sand-WaterMixture Studies)
Modified Froude Number versus Solids Concentration,Particle Diameter as Parameter (Data from Studies ofother than Sand=Water Mixtures)
Critical Velocity and the Velocity Corresponding tothe Minimum Head loss
' .. .,
-xi-
60
63
66
I'I
/Table
3.1
3.2
4.1
4.2
4.3
LIS T OF TAB LES
Title
Relative Roughness and Material Roughness Valuesfor the Three Pipe Sizes
Solid Particles Specification
Tested Combinations of Pipe Diameter, Solids ParticleDiameter, and Slope
Critical Deposit Velocity Data
Range of Parameters of the Data Reported by OtherInvestigators for Sand/Water Mixtures; Data arePlotted in Fig. 4.3
Range of Parameters of the Data Reported by OtherInvestigators for Solid/Liquid Mixtures other thanSand/Water; Data Plotted in Fig. 4.4
..
-x-
",. ,~ " .
34
40
41
47
59
62
ABSTRACT
IThe present study deals with critical deposit velocity, ''VC",
defined as the velocity at which particles begin to settle from the
carrying m~dium and form a stationary (non-moving) deposit along the
invert of the pipe. Newtonian suspensions of low solids concentra-
tions (C < 5%) are of particular interest, since the critical deposit
velocity of low-concentration mixtures is presently not well defined.
An analysis of the significant parameters in this problem
is presented and various forms of the modified Froude number are
defined and tested. ,From a regression analysis of the experimental
data, correlation of the tested parameters quantitatively defines the
modified Froude number relationship •. "
Application of the Lehigh equations to some typical trans-
port problems is examined and the economic advantages of such an
application are discussed.
-1-
1. INTRODUCTION TO THE PROBLEM
The problem investigated in this study deals with an im
portant aspect of solid-liquid transport technology in pipelines:
The critical deposit velocity, "VC". The critical deposit velocity
i,n a closed conduit separates the "non-deposit" (depo~it free) .regime
from the "deposit" regime. This velocity is sometimes also referred
to as either the.minimum transport velocity, the deposition velocity,
or just the critical velocity.
The critical deposit velocity of low concentration mixtures
(C ~ 5%) is presently not well-defined, although it is sorely needed
for application in pipeline design. Pressuri~ed sewage collection
lines, most often transporting low concentration loads, have been
shown to be economically competitive with conventional means of sewage
disposal but in need of additional design information. There exists
an exhaustive list of Newtonian slurry transport applications, which
can be found in the literature. Condo1ios et al. (1963) give the most
thorough coverage, making readily apparent the economic advantages of
pipeline transportation. Further, Shen et a1. (1970), Robinson et a1.
(1971), and Graf (1971) report the most current state-of-the-art and
econ~ic significance of the critical deposit velocity determination.
There exist generally two prerequisites in properly de
signing a solid-liquid transport system: (1) Consideration of criteria
that will ensure operation in a region of stability, and thus, provide
for safe, uninterrupted transport of solids, and (2) Minimization of
the power required to transport the solids, and optimization of system
-2-
design parameters. The critical deposit velocity relates both of
these requirements in designing a transport system which is both
economic and safe to operate~
The present study continues the investigation of the crit
ical deposit veldcity problem through the use of a modified Froude
number analysis. From a regression analysis of the Lehigh data,
correlation of the tested parameters with different modified Froude
numbers is evaluated, and equations quantifying the modified Froude
number relationship are determined. The Lehigh data are subsequently
compared with data reported in the literature. Application of the
Lehigh equations to some typical transport problems is examined, and
the economic advantages of such an application are discussed.
-3-
2. SOLIDS TRANSPORT TN PIPES-
2.1 General Remarks On Solid~Liquid Mixture Flow
It is not within the ,scope of this paper to exhaustively pre-
sent the general theory for flow of solid-liquid mixtures in pipelines.
Shen et a1. (1970a) and Graf (1971) have presented comprehensive sur-
veys on the current state-of-the-art of sediment transport in pipes,
and the interested reader is referred to these texts. However, some
general comments are appropriate as an introduction to the critical
deposit velocity problem.
Many fields of industry have become interested in the app1i-
cabi1ity of pipeline transport of solid materials along with a concern
for the related problems of solid-liquid mixture flow. In all, trans-
ported solid-liquid mixtures may vary from suspensions in water of
coal, sand, gravel, wood chips, chopped sugar cane, and ashes to
slurries of sewage sludge, polymeric solutions, and concentrated sus-
pensions. The economic advantages of hydraulic transport, the great
variety of applications, and some concepts for ,designing a hydraulic
transportation system are presented by Condo1ios et al. (1963a).
Solids 'suspensions are transported-either as "Non-Settling"
(homogeneous) mixtures or as "Settling" (heterogeneous) mixtures. The
distinction between these two classifications has been presented by
Durand (1953) and Govier et ~l. (1961). The present study is con- '
cerned with a "Settling" mixture, which exhibits Newtonian flow char-
acteristics and is analyzed as a ,two-phase flow phenomenon. The
-4-
I
susp~nsion settling characteristics in a turbulent pipef10w are not
discussed here, since the complex physics involved is beyond the scope
of this study. Reference is made to Govier et a1. (1961), Thomas
(1962), Rose et a1. (1969), or Carstens (1969,1971).
Regimes of Flow. The transport of "Settling" mixtures in
pipes is qualitatively characterized by several different regimes of
flow. Reference for an explanation of these different regimes is again
made to Shen et al.· (1970a) and Graf (1971).
The variety of flow regimes is diagramatica11y presented in
Fig. 2.1, which is a typical curve of mixture head loss versus mixture
velocity. An important distinction is made between the "Deposit"
transport regime and the "Non-Deposit" transport regime. Within the
non-deposit regime, several modes of transport prevail: .(1) p~eudo
homogeneous flow, .~ heterogeneous flow, and ® heterogeneous flow
with saltation. Flow in the deposit regime, ~, is described by bed
and dune form irregularities. Separating the deposit and the non
deposit flow regimes, ~, is the transition region identified by the
critical deposit velocity, ''Vc".
The points of division between different flow regimes is
somewhat arbitrary. Only a brief review of the flow regimes is pre
sented herein.
Pseudo-homogeneous' flow exists if suspensions of very fine
particles, with fall velocities insignificant in relation to the fluid
motion, are transported. Since homogeneity is not critically dependent
-5-
i (log)m
MixtureHead Loss
1.0
0.1
0.01
DepositRegime
~-Clear
Fluid
V (log)·mMixtureVelocity
10Non-Deposit
Regime
CD -- PSEUDO-HOMOGENEOUS FLOW; concentration gradient isnearly uniform; suspendedload transport
~ --HETEROGENEOUS FLOW; concentration gradient increases;transport by suspension and bedloads
® -- TRANSITION REGION, "Ve"; beginning of bed formation;decrease in moving concentration
@ -- DEPOSIT REGIME FLOW; bed forms (plane and dunes);eventual clogging·
Fig. 2.1: Regimes of Flow
-6-
on the flow conditions, fm (mixture flow friction factor) = f-t (liquid
flow friction factor) may be assumed. Larger particle suspensions ~y
behave similarly if transport velocities are extremely high. The
pseudo-homogeneous flow regime is characterized by a nearly uniform
vertical concentration gradient and a d~ensionless transport parameter,
CPD (see Eq • .(4-.l)), solely dependent on the relative density of the mix-
ture. O'Brien et ale (1937) and Howard (1939) investigated flow of fine
sand suspensions transported in this flow regime. Spells (1955) defines
an "equivalent true fluid" with density equal to the two-phase mixture
in the pseudo-homogeneous flow regime.
Heterogeneous flow occurs as the mixture flow velocity is de-
creased. Settling suspensions·in this flow regime will exhibit a non-
uniform concentration gradient and a noticeable increase in the mixture
pressure gradient over the clear fluid head loss curve. Particles are
transported both as bed load and suspended load now that the effect of
gravity is felt by the solids. This regime of flow is normally shown
to be the most important economically from the standpoint of total
soli~ throughput. Wilson (1942) was one of the first investigators
to present an expression for the total energy gradient for heterogeneous
flow of mixtures. Durand (1953) and his co-workers at SOGREAH developed
to date the most reliable theory of heterogeneous mixture flow trans-
port.
Some investigators separate the heterogeneous flow regime into
two: (1) transport of solids as suspended and bed loads, and (2) trans-
port of solids mainly as bed load, sliding and saltating along the
-7-
bottom of the pipe. Newitt et al. (1955) give the best account of the
reasoning for this division. It should be noted here that the distinc
tion between these two modes of heterogeneous flow is not to be mistaken
as the separation between deposit and non-deposit regimes of flow or in
no way related to the critical deposit velocity condition, as defined
in this study.
The Deposit Regime of flow is entered as the sliding bed load
of solid particles thickens and eventually becomes a non-moving bed on
the invert of the pipe. The mov~ng concentration diminishes, the clear
flow area of the pipe decreases, and flow conditions are altered. The
head loss component due to the solids is less effective, and the iln
portance of flow-through geometry becomes a governing factor in head
loss determination. Eventually, dunes will form as irregularities on
the bed surface, and plugging flow becomes a serious concern. For the
deposit regime of flow, two criteria may be employed. One is presented
by Gibert (1960) as an adapt ion of the Durand-Condolios relationship
for deposit flow conditions, and the other one is the transport-shear
intensity relationship developed by Graf et al. (1968).
A Transition Region separates the deposit and non-deposit
transport regimes. The head loss in this region-flattens to a nearly
constant value with further decrease in velocity; due to a complex
deposit-scour feedback mechanism constantly altering the relative ef
fects of the solid and liquid head loss components. The transition
region is identified by a critical deposit velocity, "VC", which is
intricately dependent on fluid,- solid, and flow pa,rameters.
-8-
/
Investigation of the transition region flow conditions and the develop-
ment of a relationship for quantitatively defining the critical deposit
veloc~ty has been the subject of many studies •. Our task is to continue
this effort.
Mixture Flow Head Loss. It has been always found seemingly
appropriate to praise the technological advancements made through the
efforts of investigators at the SOGREAH Laboratories in Grenoble,
France, namely: Durand (1953), Gibert (1960), and Condolios et ale
(1963a, b, & c). !he solid-liquid flow theory developed at SOGREAH
bas been a long-standing criteria for determining mixture flow head
loss of heterogeneous transport of solid suspensions through pipes.
An early suggestion setforth by Blatch (1906), that the mixture head
loss in a pipe is due to the clear flow head loss plus a head loss
component due to the solids in transport, was further developed by
Durand (1953) in defining a dimensionless transport parameter, ~D:
~D = (2.1)
. where i m represents the total mixture head loss; i~ the head loss due
to just the liquid phase component; and Cis the moving volumetric
solids concentration. The excess pressure gradient in this case is
often found to be proportional to the moving solids concentration.
The sediment transport parameter function is developed through
a dimensional analysis, or:
-9-
//
v acp = IL f (s -l)f (va) f (~)D -1) 1 S a gD a gd (2.2)
where (s -1) represents the relative density of the mixture, and (va/gD)s
and (v 2/gd) are, respectively, the flow and particle Froude numbers.ss
The effect of both particle characteristics and flow parameters is
evident, and the forms of~, fI
, fa' fa are determined empirically
from availab Ie data.
Further investigations of mixture flow theory and the associ-
ated economic implications were continued at SOGREAH. Later investi-
gat ions have both praised and questioned the form of the so-called
Durand-Condolios transport parameter, CPD' but not one has yet touched
on a better approach to the mixture flow problem.
im
MixtureHead Loss
-- Individual Runs
--Equi- ConcentrationLines
v , Mixture Velocitym
Fig. 2.2: Equi-Concentration Lines
-10-
// The head loss plot of a typical mixture flow run from pseudo-
homogeneous flow velocities down to deposit flow velocities was given
in Fig. 2.1. Moving concentration decreases as flow enters the deposit
regime. Determination of the minimum mixture head loss for a particular
. solids concentration flow is important in design. A rather typical plot
of constant concentration lines is shown with Fig. 2.2. Note that the
equi-concentration lines below the critical condition can only be plotted
by connecting the points of the same moving concentrations from runs with
different initial concentrations. Along these equi-concentration lines,
the mixture head loss is seen to again increase in the deposit regime.
The Vc dashed line shows the variation of critical velocity with change
in solids concentration•
. "
2.2 The Critical Deposit Velocity, "V "C....
2.2.1 Definition and Significance
The transition between deposit and non-deposit flow regimes
is identified by a "critical condition". In the present investigation,
"critical condition" is taken as the velocity at.which particles being
to settle from the flowing medium and form a stationary (non-moving)
deposit along the invert of the pipe; this will be called the critical
deposit velocity, "VC".;.
At the "critical condition" a deposit-scour feedback mechanism
transports solid particles in the form of a pulsating bed. Figure 2.3
shows typical bed motion at critical deposit velocity for plastic
-11-
suspended particles
saltating particles
particles
Fig. 2.3: Bed MGtion of Plastic Pellets in a 6-inchPipe at the Critical Deposit Velocity
pellets transported in a 6-inch pipe. Close to the pipe wall, the solid
particles are stationary. When this ,condition is observed, the critical
deposit velocity is recorded. Above this layer of stationary particles,
the remainder of the bed is sliding. Other particles shove, roll, and
saltate over the moving bed surface, and some will become completely
suspended farther from the wall. The deposit of solids on the bottom
of a pipe is a random phenomenon varying with local fluctuations of
solid and liquid parameters. Within the same pump-pipe facility, dupli-
cation of results is not easily, attainable.
The critical deposit velocity is somettmes referred to as
the Itmit deposit velocity, by Durand (1953) and Sinclair (1962), the
-12-
IIr
sediment limiting velocity, by Gibert (1960), the minimum transport
velocity, by Rose et a1. (1968), or the deposition velocity, by Wasp
et ~l. (1970). It is imperative that a clearly defined "critical
condition" becomes a primary concern in every solid-liquid transport
investigation.
When using data from other "critical condition" studies,
one must be cautious of the following: (1) Some investigators, such
as, Blatch (1906), Wilson (1942), Bruce et aL (1952), Thomas (1962),
Charles (1970), and Shen et a1. (1970b), define a minimum or economic
velocity which corresponds to the minimum head loss required for trans~
porting' a certain concentration of solids. Use of this criterion is in
accordance with how one wishes to define "critical condition". It was
found in the present and in other investigations that the critical de-
posit velocity is not in direct relationship with the minimum head loss
criterion. Implementation of the assumption that these two criteria
are identical is good only for preliminary evaluation. (2) The cri-
tical deposit velocity, approached from the non-deposit regime, is
most. often different from the critical scour velocity. To scour a
deposited bed requires usually a greater shear force, thus a higher
flow velocity, than when the same bed is deposited. (3) Some studies
define a transition velocity between sa1tating and sliding bed load
transport, which is at times mistaken for the critical deposit velo-
city.
The critical deposit velocity is an important design cri-
terion both for safe operation and for system economics, but it is
-13-
-_._-----
•I
often vaguely defined in reports of solid-liquid transport research•
Due toa lack of good definition arid reproduceability of results" it
is suggested that a conservative critical deposit velocity be used
[see also Bonnington (196l)J.
2.2.2 Previous Investigations
Interest in the "critical condition" of solid-liquid trans-
port in pipes was initiated by Blatch (1906) and continued by O'Brien
et a1. (1937), Howard (1939), and others. However, Wilson (1942)
developed the first relationship which quantitatively dealt with
parameters related to the "critical condition". As a first approxi-
mation, the total energy gradient, i , consists of a liquid component,.' m
i L, and a solids component, is, or:
. "
!Ii
I~ .,
Wilson (1942) defined both terms and obtained the following:
i' = f! va + K C (vss)m . D 2g V
(2.3)
(2.4)
Ii
~here the terms on the right represent, respectively, a liquid head
loss gradient derived from the Darcy-Weisbach equation, and a head
loss gradient due to the solids dependent on solids concentration, C,
particle settling velocity, vss
' an average velocity, V, and corre
lation parameter, K.
-14-
,:
Differentiating i with respect to V and minimizing, them
resulting "critical condition" is given as:
KCv g Dss
f(2.5)
It should be noted that the flow velocity, VC
' at "critical condition"'"
is defined here for minimum energy gradients. Nevertheless, the re-
lationship given with Eq. (2.5) relates parameters which are of im-
portance in the critical deposit velocity problem. These parameters
are: C, the solids concentration; v , the particle settling velocss
ity; D, the pipe diameter; and f, the friction factor indicating flow
resistance.
Durand (1953) used as the lower limit of his heterogeneous
flow relationship an equation defining the lUnit deposit velocity, VC'
of sand mixtures which se~arates the zones of the regUnes with and
without deposit on the pipe bottom, or:
(2.6)
The parameter, FL, known as a modified Froude number, varies with solids
concentration, C, and particle diameter, d. This is given with
Fig. 2.4a for uniformly graded material. Later, Durand et al. (1956)
report findings for non-uniform material, which is shown with Fig. 2.4b.
An appreciable difference is noted between Figs. 2.4a and 2.4b, and it
becomes questionable that these discrepancies are accounted for solely
-15-
v . 2 ...------------------,c
=15%
d [mu)
0L......J--1....-L_L.-..l.--1...-.L._L-"'--~_.L..___L.---&
o 2'.
' .. ~(a) Uniformly Graded Material
[after Durand (1953»)
32 .I
OAL.-....J..-.l-..--L.-..L---L-~--I--'----IL..--'--L.............._.L._....L.____I
o ----=----, '. .__. - ._---._---- --------
0.8
0.6
V2gD (ss-l)' 1.2
1.0
(b) Non-Uniform Material[after Durand andCondolios (1956»)
Fig. 2.4: The Modified Froude Number, FL, versus SolidsConcentration and Particle Diameter
-16-
,.I
!
II
/
by the difference in material distributions. Unfortunately~ neither
Durand et ale (1956) nor any of the later publications of the SOGREAH
staff explain this difference.
Gibert (1960) reported on and analyzed the extensive SOGREAH
data to obtain best-fit curves for Froude number, VC/~gD, plotted
against solids concentration, C. Subsequent to the study of Gibert
* .(1960) , Graf et al. (1970) included the effect of relative density,
given by J2(s -1), - as was similarly done by Durand (1953) - ands
Gibert's best-fit curves were replotted and are given with Fig. 2 0 5.
This figure shows the general trend of results to be remarkably in-
variant for sand and gravel of particle sizes d ~ 0.37 mm. The curve
for this larger material can be thought of as being a maximum envelope
of FL-values. For finer materials, in the range of d = 0.20 mm and
less, there are distinctive variations in the curves. Condolios et a1.
(1963b) report a figure similar to Fig. 2.5 but only include an envelope
curve for graded and mixed sands of d > 0.44 mm. Figure 2.6 is a re-
plot of Fig. 2.5. It should be noted that Fig. 2.6 conforms closely
to the non~uniformmaterial results reported by Durand et ale (1956)
in Fig. 2.4b. It is expected(!) that both Gibert (1960) and Durand
et ale (1956) used the same set of SOGREAH data. Furthermore, it is
believed that Figs. 2.4band 2.6 supersede Fig. 2.4a; the latter is
a result of earlier SOGREAH studies.
*"TransJation and evaluation of Gibert (1960) was undertaken byOner Yucel, Lehigh University.
-17-
2.0 Vc/2gD (ss-l)I
1..5
--.'~-----~ -~- ----- ~--~. --~._~
Adopted from Gibert (1960)
____ Sand of d > 0037 nun'
___ Sand of d = 0020 nun
-1.0
/
0.5
--.------~--
..... -.,.
---
Fig. 2.5: Modified Froude Number versus Concentra1:ion.;Particle Diameter as Parameter
o 2.5 5.0 '10.0
,, ~
;, ~
2.0 VcV2gD (s _1)'i s
1.5
Equi-Concentrations Adopted .from Gibert (1960)
--- - ------- - --- ----- - -- - -----
·1.0 '
0.5
d [nun]
Modified Froude Number versus Particle Diameter;Concentration as Parameter
o ._._,._ _ . ..0.5 _.__~--.-._. _Fig. 2.6:
1.0 1.5
-18-
General agreement with the relation, as defined in Eq. (2.6)
and plotted in Figs. 2.6 and 2.4b, are found throughout the literature.
Figure 2.4b is recommended by Graf (1971).
Gibert (1960) also discussed a theoretical approach to the
·critical deposit velocity problem, considering the "critical conditions"
of flow in a conduit irregardless of flow-through geometry, to be re-
1ated through the Froude Law of similitude. A discussion of Gibert's
analysis is found in Robinson et al." (1971).
Sinclair (1962) conducted tests on sand-water, iron-kerosene,
and coal-water mixtures at concentrations up to 20% flowing in 0.5-inch,
O.75-inch, and 1.00-inch pipe. Through a dimensional analysis of the
variables expected to significantly influence the critical deposit
velocity, Sinclair (1962)" arrives at an equation, such as:
Vmax f [dSS]
s D(2.7)
where the modified Froude number is expressed with a solid's particle
diameter, ds5
." He observed that the critical deposit velocity reaches
a maximum between 5 and 20% solids concentration, so that the effect
of concentration could be eliminated by using Vmax instead of VCO
Sinclair (1962) wrote Eq. (2.7), for d > 1.5 mm (when C does not enter
the problem), as:
(2.8)
This may be compared with Durand's results, similarly expressed by:
-19-
(2.9)
For smaller particle sizes, Sinclair (1962) examines the
relevance of boundary layer theory to the problem, and suggests that
particle diameter, d , takes precedent over the pipe diameter, D, in85
their relative influence on the modified Froude number. It is within
this smaller range of particle sizes that the present study is con-
ducted.
Shen et al. (1970b) and others attempt to correlate critical
deposit velocity with other important parameters in the form:
(2.10)
The exponents, a, b, c, and d, and particularly the coefficient ~;.
vary greatly, as could be expected, from one study to the next. The
form of this function is questioned because of its inhomogeneity and
is to be used only with extreme caution in data correlation.
F~OW and particle Reynolds numbers have been investigated
for their applicability as criterion in the critical deposit velocity
problem. Spells (1955), Charles (1970), and studies by Cairns et al.,
as reported by Sinclair (1962), correlate the Reynolds number with a
modified Froude number relationship. Correlation in these studies,
however, is related to the minimum energy gradient criterion.
A modified Froude number relationship apparently presents a
rather good criterion for evaluation of solid-liquid mixture flow
-20-
(;0),60
through pipes. Its relationship to other parameters significant in
the critical deposit velocity problem will be re-examined in the pre-
sent study, and experimental findings checked against the SOGREAH data.
2.2.3 A Modified Froude Number Analysis
When transporting a solid-liquid miXture through a closed
conduit, one may expect the following variables to be of importance:
(1) Flow Parameters -
V, miXture flow velocityg, gravitational acceleration
vss ' particle settling velocity
(2) Fluid Parameters -
P, carrying fluid densityv, kinematic fluid viscosity
(~) Pipe Parameters -
D, pipe diameter€, pipe roughness
tan a, pipe slope
(4) Sediment Parameters -
Ps ' solids particle density
d, mean particle diameterf , particle shape; sphericitys
non-uniformity coefficient of graindistribution
C, moving volumetric solids concentration
Proper grouping of variables into dimensionless parameters
was reported in Graf et ale (1970) and is re-examined here:
f [~, (s -1) , VD d € tan a,dgo
cJ (2.11)v' D'f ' D' = 0s %0,
gD s
The relative density~ (s -1), comes from (ps-p)/p where s = P /p.s s s
-21-
I
It is expected that the flow Reynolds number, VD/~, does
not playa significant role in this problem, and it is omitted from
the analysis without loss of generality. The mixture flow velocity,
V, and pipe diameter, D, are accounted for by the remaining parameters
in the relation, Eq. (2.11). The kinematic viscosity, ~, which depends
on temperature, for all practical purposes varies insignificantly.
Further, a Reynolds number near the critical deposit velocity is very
unstable, because the flow-through geometry, D = 4~, varies con
tinuously with fluctuating solids concentration, along with changing
. clear flow-through velocity.
Replacing the general flow velocity, V, with the critical
deposit velocity, VC' and considering the particle shape factor to
be unity for natural quartz grains or already included in the adjust-
ment of non-spherical particle sizes, Eq. (2.11) is rearranged and
lI
i!
IJ
given by:
d eD' D'
~otan S, <\so ' cJ = 0 (2.12)
Note that the flow Froude number, v/~, and the relative density,
(s-l), both given in Eq. (2.11), were combined in a densimetric or. s .
modified Froude number, Vc/j2gD (ss-l). Equation (2.12) is somewhat
similar to relations proposed by Durand (1953), Sinclair (1962), and
Barr et ale (1968).
For· a certain relative pipe material roughness, e/D, and
solids grain size distribution,.dso/dso ' the applicability of
. Eq. (2.12) will be tested in the form of:
-22";
/
I2gD (s -1)
s
~ [tan e]
(a)
d- or dD
Vc-----=:..-.-- ~ [tan eJ2gD (s -1). s d
D or d
(b)
c
Fig. 2.7: Plot of Equation (2.13); the Modified Froude NumberRelationship
. -.(2.13)
Equation (2.13) is displayed on plots such as given in Figs. (2.7a)
and (2.7b). The effect of pipe slope, tan e, is not a major concern
in this study. The left side of Eq. (2.13) will absorb the tan eargu-.
ment, and the best trigonometric re1at~onship will be determined after
fitting data against both:
and, .
-23-
The left side of-Eq. (2.13) is a modified Froude number. The form
of this parameter, raising both D and (s -1) to the 1/2 power, hass
been tested and shown to be a reliable criterion.
It is felt t;hat without loss of generality, it may become
frequently important to replace the relative particle to pipe diameter,
d/D, by the particle diameter, d, itself. In this instance, the signif-
icance ofD is considered to be wholly described in the Froude number.
Sinclair (1962) remarks that when the particle is such a size that it
is wholly immersed in the region where viscous forces predominate, as
our sand particles are, d/D does not enter the correlation.
Investigators, like Bruce et a1. (1952), Govier et ale (1961),
Thomas (1962), and R~se et ale (1969), consider slip between the solid
and liquid phases, v /V or V /V (referred to as ''hold-up''), to be ass s
parameter (jf maj or importance. This concept requires a thorough ..
treatment of particle dynamics, beyond the scope of the present study.
It is therefore considered that near the critical deposit velocity,
particles have already settled into a sliding bed; consequently, only
the size and moving concentration of particles are significant.
In the subsequent discussion, data will be presented and com-
pared in theTjlay su~gestedwith Fig. 2.7a.
-24-
3.. LEHIGH EXPERIMENTS
The experimental facility consists of: (1) a vari-drive
motor-pump assemblage, (2) an adequately flexible pipeline arrangement,
(3) a sediment feed and removal system, and (4) the necessary measur
ing and regulatory devices. Figure 3.1 schematically illustrates the
general scale of the overall system. Detailed features of the sedi
ment handling equipment are provided in Fig. 3.4.
Vari-Drive Motor-Pump. The hydraulic horsepower was supplied
from a vari-drive motor-pump assemblage, functioning as the heart of
the system. The pump, furnished by Ellicott, is a single suction
centrifugal type with cast bronze casing and impeller. The suction
pipe is 5-1/2 inch 1.0., discharge pipe is 4-1/2 inch 1.0., and the
impeller diameter is 13-5/8 inch 0.0. During the operation of the
pump, cooling water is added continuously to the seal on the motor
side of the pump, also providing a lubricating interface.
The drive unit is a Westinghous~ - 3 phase 60 cycle 125 Hp
'~gna Flow" motor and is regulated by a vari-drive control. The
driving unit is of the integral type, is water cooled, and has an ad
justable speed range from 100 to 2153 rpm. Along with the motor,
-25-
,.
",.c:'--
6" CouplingSleeve
3" Loop ~\OO"System o~'i ~\o~ ~
~~_~\e~~ ,\o~~_vC:Je~ .
.Plexiglas- ~e~\
.,..LObservation~~,--Section
1151
a" DishargeRate Control
'Gate Valve
6" Foxboro MagneticFlowmeter
125 H. P. VaridriveMotor- Pump Assemblage
Water
6 11Flush Valve to
Collection Sump
Fig. 3.1': Solid-Liquid -Transport Test System
-26-
//
there is an operator's station, excitation unit, and a type 5L Auto
starter. The entire system operates on 208 volts AC.
The pump and vari-drive motor assembly survived 18 months
of testing. Pumping efficiency and impeller capacity were not notice
ably altered throughout the testing period. Sand mixtures presented
no p~ping difficulty, however, the 3.63 mm diameter plastic pellets
were extruded apparently along the surface between the impeller and
encasing seal. Resulting conglomerations of plastic strands within
the pump would put a strain "on the motor at low flowrates, causing
sudden velocity fluctuations. This complication is explained further
in Section 3.3.2.
Pipeline. From the pump, mixture flow is discharged through
a 6-inch Foxboro Magnetic Flowmeter leading to a horizontal reach of
8-inch pipe. An 8-inch gate 'valve regulates pump discharge below
flawrates of 200 gpm. Often times the partially closed valve would
cause difficulty in establishing stable flow conditions when critical
flawrates occurred in this lower flow ~ange. The solid-liquid mixture
is then lifted to the test-floor elevation in 6-inch pipe.
Along the test length of approximately 40 ft, measurements
are obtained, pipe slope is adjustable, and mixture flow phenomena
are visually observed. A 4-inch pipe was installed together with its
Plexiglas observation section; subsequently, a 6-inch pipe and
Plexiglas section were installed. A strobotac set at a high frequency
response aided the observation of solids flowing through the Plexiglas
section, such that an accurate description of flow regime was
-27-
obtainable. For example, Fig. 3.3 pictures the prdgressive dune trans
port of sand particles in the deposit regime, as seen through the
6-inch observation section. Both pipe sizes and slopes were altered
throughout the testing program in accordance with the investigation of
variable parameter affects. Figure 3.2 shows the horizontal 4-inch
diameter pipe s.etup.
A "Loop System" follows which is employed as a device for
s~ult.aneously~asuring mi~ture flowrate and solids concentration.
Locat~d~~top the balcony-floor elevation between the 3-inch vertical
pipe sections, commonly referred to as the "Riser" and "Downcomer''', is
the main air-release for the system.
The flow, upon leaving the "Loop System", bypasses a closed
3-inch sediment flush valve. and enters a 6-inch vertical pipe, where
sediment is gravitationally fed when an increase in concentration is
desired. Flow continues downward to where a 6-inch gate valve empties
the system and a 2-inch pipeline connects the city water supply. The
system pressure was maintained and water supply assured through use
of a constant pressure control valve (A in Fig. 3.1) set at 20 psi
on the 2-inch supply line. A 2-inch check valve (B in Fig. 3.1) pre
vented backf10w to the city supply under excessive system pressures.
The circuit is completed with 5-1/2 inch pipe leading to the
suction side of the pump.
The pipeline, secured both laterally and from hanging steel
supports, could safeiy handle flowrates up to 1000 gpm. Wear on the
-28-
e.-
.,r:.........._ ..__
Fig. 3.2: Setup for Tests in a Horizontal 4-in.<iA...Diameter Galvanized Pipe
Fig. 3.3: Low Flow Dune Transport of Coarse Sand Particlesin the Deposit Regime
-29-
inside pipe finish was apparent, however, not of serious consequence.
Due to old pipe sections, iron oxide coloration eventually became a
persistant recurrence causing only some difficulty in flow visuali
zation. The system water was flushed clean when flowrates were lowered
to a range ensuring no sediment transport. Transitions were attacked
by the sand, but the use of tee fittings in the critical location of
90° elbows saved the necessity of replacement. The most persistent
problem was caused by sand particles jamming the gate valves. Other
valves on the market would have gauranteed greater success.
Pipe lengths and fittings were supplied by the Bethlehem
division of Hajoca Corporation, and the Fritz Laboratory .machine shop
handled material alterations.
Sediment Feed and Removal System. The sediment feeding
apparatus underwent several adaptions, until the technique, as ex
plained here and illustrated in Fig. 3.4, was successfully applied.
Supply valve 2 and overflow valve 3 are opened as the mixing chamber,
isolated from the system by the closed miXing valve 1, is filled with
solids material. Water is displaced through the overflow line as the
mixing chamber is filled. Valves 2 and 3 are then closed and valve 1
is opened, fluidizing the solids and gradually feeding the particles
into the flowing medium.
Also illustrated in Fig. 3.4 is a sediment removal facility
(employed as a time-saving technique) for removing the solids or un
desirable foreign material from the system and preventing discharge
-30-
Sediment Feed
SupplyValve 2
MixingValve I
3 II Downcomer
.Overflow t
3 II Sed iment
Flush Valve
Clear Water
to Sump
Sediment
Separation Device
~ig. 3.4: Sediment Feed and Removal Facility
-31-
of polluted water to the collection sump. The 3-inch sediment flush
valve was opened enough to maintain positive pressure in the system
and divert the mixture flow into the receiving chamber of the sedi-
ment'separation device. Two square feet of No. 60 cooper mesh screen-
ing prevented flow through of solids material. The screened clear water
was removed to the sump.
Sediment feeding was the more troublesome of the two oper-
ations. Both the mixing and supply valves were replaced because of
jamming, which caused unexpected backup of sand slurry from the mixing
chamber.
Measurement and Flow Regulation. The volumetric concentra-
tions of solids and the mixture flowrates were determined from "Loop
System" head loss readings. Arrows land 2 on Figure 3.1 indicate the
respective locations of "Downcomer" and "Riser" pressure taps, both
with 1.50 m (=59.1 in.) head loss lengths.
Loop readings were repeatedly checked against flow recordings
from a Foxboro Magnetic Flowmeter by means of a DYnalog Receiver measur-
ing accuracy to within 1 percent of· full scale, throughout the scale
(approximately ±25 gpm). A Prandtl tube (C in Fig. 3.1) was employed
to verify both the "Loop System" and flowmeter measurements of mixture
velocities. A Pitot tube sediment-sampling device (D in Fig. 3 •.1)
checked the "Loop System" indication of solids concentrations. Further
discusslon on determining concentrations and flowrates is .found in
Section 3.2.
-32-
Two Venturimeters were investigated for their applicability
as mixture flow measuring devices, the results of which are reported
by Robinson et a1. (1970). A new 3 x 2 inch Venturimeter(E in
Fig. 3.1) and an antiquated 4 x 2 inch device (F in Fig. 3.1) were
tested and later used in checking flow conditions for this particular
study.
The mixture head loss length for the test section was 3.60 m
(=141.8 in.), as located at the arrows marked 3. At each pressure
tap location, four holes, 3/32 inch in diameter, were drilled diagonal
ly opposite about the circumference of the pipe. Brass fittings were
assembled and connected with poly-flo tubing for transmitting the hy
draulic pressure. Manometer fluids were selected according to the
required range of readings. Most often air-water readings were adequate,
however, a 2.95 fluid-water medium was needed at extreme flow conditions.
The 50.0 in. manometer scales were graduated in tenths of an inch, read
ings to a hundredth of an inch were estimated, and each reading was con
verted to feet of water column. Minor manometer fluctuations always
existed, partly due to the uneven distribution of sediment concentration
through the large system arid also due to the effect that concentrated
slugs of sediment had on the pump's capacity for maintaining a constant
mixture flowrate.
Flowrates between 200 and 1000 gpm were regulated by a vari
drive rheostat control, located at the operator's station. The 8-inch
discharge valve controlled lower range f1owrates. Sediment feed rates
were not rigorously monitored, except for an attempt to evenly distri
bute the sediment throughout the system.
-33-
3.2 Measuring Techniques
Clear-water calibration of the system was the initial course
of action. The "Loop System" head loss readings were then evaluated
and checked against floWmeter, Prandtl tube, and Pitot tube measure-
ments.
3.2.1 Clear-Water Tests
Tests of clear-water flow were conducted to determine material
roughness characteristics of the 3-inch '.'Loop System" pipes and the 4
and 6-inch diameter test lengths. Friction factors, f, were calculated
from the Darcy-Weisbach equation, evaluating manometer head loss read-
ings and Prandtl tube indication of velocities over the ranges of
Reynolds number indicated in Table 3.1. Also summarized are the
Pipe Specification e/D e Reynolds Nos.(ft)
Loop System:
3 in'. ¢ commercial steel 0.00004 0.00001 2.48 x 105 to4.77 X 105
Test Length:
4 in. ¢ galvanized 0.00009 0.00003 1.97 x 105 to3.58 X 105
6 in. s6bla~k steel 0.00032 0.00016 1.39 x 105 to3.76 X 105
,
Table 3.1: Relative Roughness and Material RoughnessValues for the Three Pipe Sizes.
respective relative roughness values, e/D, and material values, e,
determined from the Moody-Stanton Diagram of friction factors for
commercial pipe. The friction factors for all three pipes falL in
the transition regime. For further determination of friction factors
-34-
., /
I)
at any mixture flow Reynolds number, an explicit solution of the
Colebrook-White equation was used•. Evaluation of extensive "Loop
System" data required this type of solution for f
3.2.2 The Loop System
The "Loop System" developed by Einstein et a1. (1966) was
used to simultaneously determine the mixture f1owrat~, ~, and the
solid phase concentration, C. The device consists of two identical
vertical pipe sections with oppos·ite flow direction. Pressure head
differences are obtained over these vertical pipe sections, namely,
the ''Riser'' and the "Downcomer" section. The head loss in the riser
section is
+ ~!, (;)"J D 2g (3.1)
and in the downcomer
(~)a.= -LIe (s -1) + ~1~ [1
. D s . J D 2g (3.2)
)
where L represents the head loss length in either section, CR
and CD
are the solids concentrations in the riser and downcomer pipes, and
~ is the total mixture f1owrate.
If the summation. and the difference of Eqs. (3.1) and (3.2) are
respectively computed, the resulting equations are
-35-
./
1.0 Fig. 3.5: Loop System Charts
0.5 Qm (cfs) 1.0
. -,
Chart 2
I-C= 10%
~
C= 5%, ..
fo-
e= 1%
I I
o
c.c.<] 10I
a:.c.<]
-
-a::" ~
<{I
3:LL 2()oZ
.c':
, Qm (cfs)
0.5o
For Type #0 Sand:125
d=0.88mm
5s = 2.65
- T=70°F0::wlOO~<!::lJ..0
I . 75w z..... Chart, II -,C
.c.'<]
+ 50a:
.c.<]
25
~ + lUtn2L = (5 -1)
s
v. AssQ
fC(1-C)2 + 'Y
e[1 + (s -l)C]s (3.3)
(3.4)
The fluid f10wrate, Qf' in Eqs. (3.3) and (3.4) had replaced the total
f1owrate, ~, to dist~nquish between solid and liquid phase flowrates,
or Qf = ~/(l-C). C is the average volumetric concentration of solids
if flowing through a horizontal section. The symbol 'Y represents ae
pressure gradient for mixture flowrate, as
12g (3.5)
It is seen that knowing riser and downcomer head loss read-
iugs for a solid-liquid miXture flow, solids concentration, C, and
mixture f1owrate, ~, may be obtained from Eqs. (3.3) and (3.4).
To expedient the determination of ~ and C from loop head
loss readings obtained while testing, a program was developed and ex-
ecuted on the University's cnt 6400 Computer to print out data for
plotting two charts. Plotted output for coarse sand particles at. .
70°F is illustrated in Charts 1 and! of Fig. 3.5. A (6~-6hn) cor
rection curve shown below Chart 2 was determined from clear-water
evaluation of the riser and downcomer readings. A set of charts were
plotted for each of the three types of particles· investigated, using
re?dings determined from two different system temperatures of 70°F
-36-
and 90oF•. The program calculated relative values of ~hR and ~hD in
functional relationship with various input combinations of ~ and C.
~ and C were generated in 0.10 cfs and 1% increments, respectively,
and up to 2.15 cfs and 20%. The friction factors for each Reynolds
flow number were explicitly determined from an equation developed by
Wood (1966):
f= a + b~-ce
(3.6)
,
which is a best fit solution to the Colebrook-White relationship. a,
band c are simple power functions of e/D, e/D determined to be
0.00004 for the 3-inch loop. pipes.
Appendix A i11ust~ates, by means of an example, how concen-
ttation and mixture f10wrate for a particular test run are readily
determined from location of head loss readings on Charts land 1. Ap-
plication of the c1ear-water correction data is also examined.
Loop indications of C and ~ were checked against Prandt1
tube and Pitot tube measurements and adjustment of the loop data
recommended. However, it was found that adjustment is only necessary
for data,in the heterogeneous flow regime. The method of evaluating
the loop data with respect to Prandtl tube and Pitot tube findings is
explained in Appendix A.
-38-
3.3 Description of Experiments
3.3.1 Range of Parameters Tested
The important parameters in the critical deposit velocity
problem were identified in Section 2. To understand the interrela-
tionships involved, it is paramount to st~dy the different effects
due to independent variation of. each parameter. Herein is described
the attempt at satisfying that requirement and a qualification of the
extensive data compilation•.
A 4-inch and a 6-inch diameter pipe, each one having adif-
ferent pipe roughness, as shown in Table 3.1, were evaluated for their
relative effects on Ve• Each was tested separately at different slopes,
assuring always a sufficient upstream flow transition length. Most of
the data were obtained with the test section placed in a horizontal
position. Some data were also obtained for both a positively and neg-
atively sloped alignment, in the hope of showing some indiction of the
tan e variable effect on critical velocity determination. The positive
slope tested was tan e = +0.027, and the negative slope, tan e = -0.060
(geometrically speaking).
Three types of solid particles, wholly described in Taqle
3.2 and pictured in Fig. 3.6, were tested in various combinations with
D and tan e variables, as are listed in Table 3.3. The mean sand
*particle diameters and non-uniformity coeffieients, ~o and dg9/~O
*dgo/~o was selected for indication of non-uniform grain distributionto expedient the compilation of data s~ilarly reported by other investigators. In a normal Gaussian distribution, it is often shown thata 95% confidence interval is represented by the elso and ~o particlesizes. This adequately characterizes the particle aggradation.
-39-
3.3 Description of Experiments
3.3.1 Range of Parameters Tested
The important parameters in the critical deposit velocity
problem were identified in Section 2. To understand the interrela-
tionships involved, it is paramount to study the different effects
due to independent variation of each parameter. Herein is described
the attempt at satisfying that requirement and a qualification of the
extensive data compilation.
A 4-inch and a 6-inch diameter pipe, each one having a dif-
ferent pipe roughness, as shown in Table 3.1, were evaluated for their
relative effects on Ve. Each was tested separately at different slopes,
assuring always a sufficient upstream flow transition length. Most of
the data were obtained with the test section placed in a horizontal
position. Some data were also obtained for both a positively and neg-
atively sloped alignment, in the hope of showing some indiction of the
tan e variable· effect on critical velocity determination. The positive
slo?e tested was tan e = +0.027, and the negative slope, tan e = -0.060
(geometrically speaking).
Three types of solid particles, wholly described in Table
3.2 and pictured in Fig. 3.6, were tested in various combinations with
D and tan e variables, as are listed in Table 3.3. The mean sand
. *particle diameters and non-uniformity coeffieients, ~o and dgo/~o .
*dgo/%o was selected for indication of non-uniform grain distributionto expedient the compilation of data similarly reported by other investigators. In a normal Gaussian distribution, it is often shown thata 95% confidence interval is represented by the els o and ~o particlesizes. This adequately characterizes the particle aggradation.
-39-
Solids Material d50 ~o/%o s vs ss(mm) (ft/sec)
Quartz Sand:
4,0 0.88 1.21 2.65 0.312#00 0.45 1.07 2.65 0.189
Plastic Pellets:
PP 3.63 -- 1.38 0.697
Table 3.2: Solid Particles Specification
---c-- - -------
(a) Sand 4,0
em 11II1 11111 t1111 1111 1"·12(c) Plastic Pellets
'i's = 0.795
-40-
(b) Sand #00
Fig. 3.6: Photographic Representation of the Three Typesof Solid Particles Investigated; (a) Coarse Sand Particles, (b) Fine Sand Particles,and (c) Plastic Pellets
respectively, were determined from a standard sieving analysis and
remained constant throughout the testing period. The highly-silica,
Pipe Diameter, Din. Mean Particle Diameter, Pipe Slope,(Material Roughness, % tan e
eft) (Specific-G~avity, s )s
4 6 0.88 0.45 . 3.63 0 -0.060 0.027(0.00003) (0.00016) (2.65) (2.65) (1.38)
* * ** * ** * ** * *
* * ** * ** * *
..
* * ** * *
Table 3.3: Tested Combinations of Pipe Diameter,Solid's Particle Diameter, and Slope
quite uniform, quartz sand was supplied by Whitehead Brothers; Co. in
New Jersey, and the plastic pellets were manufactured by B. F. Goodrich
Co. in Ohio.
The·effect of particle shape or true sphericity, ~s' is con
sidered in adjusting the apparent mean particle size of the plastic
pellets by the equation:
(%0)effective= (dso)apparent
~s(3.7)
-41-
'f i-s defined as the ratio of the surface area of the equivalent-volumes
sphere to the actual surface area. It is an isoperimetric property of
particles, and its hydrodynamic influence on settling velocity is
developed by Graf et al. (1966).
The cube-shaped plastic pellets, with average dimensions of
1/8 in. x 1/8 in. x 3/32 in., indicate an "apparent" particle diameter,
d50
= 2.89 mm. Upon application of the cube-shape sphericity factor,
'f = 0.795, Eq. (3.7) defines an "effective" particle diameter,s
~o = 3.63 mm. Irregular pellet shapes were removed, but a distri-
bution was not determined.
The respective settling velocities were found from a graph
and equation presented after Budryck by Durand (1953, p. 100).
Budryck's graph and equation cover the entire range of settling velo-
cities for "quartz grains" of 2.65 specific gravity in a quiescent
medium. The consideration of sand particle sphericity was not neces-
sary. Plastic pellet settling velocity, however, was determined from
the "effective" particle diameter.
The specific weights of the solids, s , were provided by thes
material suppliers and are listed in Table 3.2.
Volumetric concentrations of 0.1% < C < 17% were handled
at flowrates ranging from 0.1 cfs (~50 gpm) < ~ < 1.8 cfs (~OO gpm).
The system temperature was recorded for each test run and sometimes
varied from 60°F < ~ < 100°F. The effect of temperature on the loop
re~dings was accounted for, as explained in Section 3.2.2.
-42-
3.3.2 Testing Procedure
Preparation for a Series test run involves selection of a
pipe diameter, D, (with determined material roughness, e); the adjust-
ment of the pipe slope, tan e; and the feed of solid particles, d60 '
(represented by solid's specific gravity, s , and. a non-uniformity. s
coefficient, ~o /%0) into the system.
For a particular test series, the solids are circulated in
a nearly pseudohomogeneous flow condition which ensures uniform distri-
bution of the particles throughout the system. Once conditions were
stabilized, the f10wrate, the moving solids concentration, and the
test section head loss readings were recorded; these are compiled in
Appendix B. A qualitative description of the mixture flow, as observed
through the Plexiglas section, is thereon commented. F10w~ates are
then decreased to the heterogeneous flow regime, and there becomes
noticeable a not so unexpected development. The moving solids concen-
tration diminishes, due to the premature settling of particles in the
larger 8-inch pipe, located upstream from the test section, exhibiting
a transport flow capacity less than that within the 4-inch or 6-inch
test sections.
Further decrease in f10wrate produces heavy bedload tr~nsport
in which most particles are either rapidly sliding along the. invert or
sa1tating into the clear flow area of the pipe. Subsequent f10wrate
changes are more finely incremented. Lowering the flowrate to ave10-
city at which the bedload begins pulsating between deposit and
-43-
// non-deposit flow conditions, the sliding bed thickness builds and
there exists no measureable transport of the bedload particles. In
this study, this is the definition of the critical deposit velocity,
Vc• The solids concentration corresponding to that particular Vc is
. recorded just prior to the critical condition, when all particles are
in transit.
Readings are also recorded in the deposit regime to complete
the data required for plotting the associated head loss curves. Dune
~ormation and dune transportation are an ever fascinating phenomenon
at these low flow ranges. Clogging of the system was never encountered.
In the early stages of this study, runs were repeated to check
the consistency of data measu~ement. Once satisfactory agreement was·
obtained, solids were added or removed to change the concentration. At
critical conditions, the concentrations never exceeded 7% by volume.
Inconsistencies are experienced in any sediment transport
study, but low concentrations in this study presented an unusual prob-
lem. The necessity of almost fully closing the 8-inch flow discharge
valve for reaching low critical velocities induced local scouring of
the already well-deposited bed in the 8-inch pip~. Sudden slugs of
sediment would then deposit in the test section at one moment, and
completely scour clean the next, under the same flow conditions. The
transport of plastic pellets.posed an additional difficulty. Low flow
conditions did not sufficiently entrain the pellets to flow freely
through the pump. Rather, particles slid down between the seal and
-44-
the impeller, straining the motor and causing sudden variation in flow-
rates.
After several runs were made at a variety of concentrations,
the data were plotted on a typical mixture head loss versus mixture
velocity graph, as explained in Appendix B, and one of the parameters
changed for subsequent tests.
-45-
/ 4. EVALUATION OF EXPERll1ENTAL DATA
4.1 Analysis of Lehigh Results
Nine series of tests were conducted to determine the critical
deposit velocities for varied concentrations of sand and plastic pellets
transported with water in a pipeline. Most data were recorded from
sand-water tests in a horizontal pipe over a range of low solids con-
centration (C < 7%). -It is expected that within this lower range of
solids concentration, both the particle diameter, d, and solids con-
centration, C, effect the critical deposit velocity value.
By testing various combinations of solids concentrations, C,
particle diameter, d, specific weight of solids, s , pipe diameter, D,s
and pipe slope, tan e, different critical deposit velocities were re-.
corded and compared. All experimental data are first tabulated and
then plotted as mixture head loss against mixture velocity (see Ap-
pendix B).
Critical Deposit Velocities. The critical deposit Veloc-
ity data are summarized in Table 4.1 with indication of run numbers-
for each series of tests, the volumetric solids concentrations, the
critical deposit velocities, and four modified Froude numbers. These
four modified Froude numbers are defined in Table 4.1 and were computed
for each critical deposit velocity. Froude number (I) is the modified
form, after Durand (1953), for critical deposit velocities in hori-
zontal pipeflow. Subsequently, both Froude numbers (II) and (III) are
introduced to evaluate critical deposit velocities in sloping pipes as
well. Froude number (IV) is suggested by Wasp et ale (1970).
-46-
--"--_._--_.-----
----- --------
-----_._-_._..__._--_._----_._---------~----------_.. ._-.._--_. -'._._~ ._--- -_._._~---_.,--------------
MODIFIED FROUDE NUMBERS EVALUATED •••
Vcp.. (I) = -,=.===::;- .... - ....-..-----. ----------r ,._J:~D._~S.S -1)' ._. u •__._~ __~~ ~ ••
Vc--------------.---------------- F .. (II) - = -[1 - tan eJ. . r. JZgD (s -I)' .___ . s .. . _
V~----~~-------.----.. F (III) = C -- . --.~- --~-_.-~~~
r .IZgD (s -1) [1 + tan eJ \______ V__ __ _ s. _ _ _ _ _ _
------ - ..... _--_.._._---________________ R VOLU;~ETRIC CRITICAL _.. _ _ .MOOIFIED _... . _
U SOLIDS DEPOSIT FROUOE. .________ N CONCENTRA TI ON VELOCITY. NUMBER .._
t---+--------t-------t- (I>--<II>-CI I I>--<IV)-_______ ~__._. ._ ..__ . (PERCENT) CFT/SEC)
1.6872.0052.1041.966
- ...._--~- .50._-'- ft.80 __.807 .855 .832__ • __ 4
"'... .-.-. .~ ------1.00 5.10 .857 .909 .884
.._---- 3.00 ----- 5.35 .899 .953 .928"-._- .__.- -- - ._-- . ----
7.00 5.00 .841 .891 .867
--=--=~~= ••. _seri~S- G-Ol { mrm~~~ ~~:::~0~o·f~. MM__ ~-~--===~=-= .------------·I---:.--I----~-·-·-=--r_·------···--- ------ ---- - --_._-~--._~...,........-:-~
6 .12 3.90 .65~ .656 .656 1.4477 .15 4.65 .782 .782 .7821.725__ •__ ~ •._. _-0_' _ __ ._. _0__ .. ~__ ._ ._
6 .20 5.10 .857 .857 .857 1.6929 .50 5.35 .899 .899 .899 1.985-------_._---- _._--- ._._---_._- .__.. _. ----... -' --- .. ---- .._.1 .50 5.00 .841 .841 .841 1.855
~ 1-10--- .60 ~ .. 80__._ .975 .975.975 ?.15t.11 1.006.40 1.076 1.076 1.076 2.374
__. .__ 2 l.OO .. .5.50 .. __ .925: .925 __ .925_2.0403 1.75 5.75 .967 .967 .967 2.1334 __ __2. 0a __ .. _ ___ 5 • 7 5 ____ • 96 7 • 967 • 96 7 _2. 13 35. 5.00 5.95 1.000 i~ooo 1.nOn 2.2n7
--------_.- ----- --_._-~.,._~_........_-~_.~_.- ~~-_.~.~-----------
==~~== ·.~erI~~G~02-{ mn~~d~;~:~:~o~o· ~~:HH_~ ~ _
___~_~ _1_ .._2
_____________._.3__4
-_...•__.- .~_._-~_.._----Table 4.1: Critical Deposit Velocity Data
------.. ---47- -_._-----,..
._--------------------
________________ ._R. VOLUMETRIC CRITICAL MODIFIEDU SOLIDS IJEPOSIT FROUOE
_________ . _N CONCENT~ATION VELOCITY _ . NUt1BER .t--4------~-----+_-(n-(I1)-(II I) - (IV)-
-------------.-- (PERCE NT) (FT/SEC)
._- -- _._--_._- .._--- - { ._. - - .. .
PARTICLE DIAMETER::: .98 MM_._ Series BS-OL 'PIPE DIAMETER::: E). 00 IN.
PI~E SLOPE = 0.000_._-------- .-_._--- _.. __ ..._-_._--_._-_.
1_~ 2
3
------------ --~ --
~ .80 5.401 • 1 0 6. 70 .3.00 7.25
____? 00. . J:~40
.376
.920
.99~
1_.016
---_._--- . ----,~..~-~~-.876 .878 2.074
.• 920 .920 2.171.995 .995 2.349
1.016 . __ 1. 016 2_~3_98 _
2.0182.3_962.475
.6671.0291.063
.8551.1H51.048
----._-------
___-:: =~~=:::~~es -:S-03-{mrm~i~ ;~E::ir ~0·~~.HH:-:-_:=-=-__ e-- _
-------.--- -- --i--·:-t~- ·-~-F_~~~i -_~_:i:m- ' -------------~---
{
PARTICLE DIAMETER::: .45 tiM . _Series G-OOl PIoE. DIAHETER. ::: 4.00 I.N. .
--....,.------1---....------- _..£l_IPE_SLOPE_= Q.!.9 0_0 .. -'-- t _
------- -_..5___ __ ______!O 5____._--~._-
2.75__ ----.462 .462 .462 1._141 ---------6 .10 4.10 .683 .689 .589 1.7017 .• 20 4.80 .807 .807 .807 1.991----- ---_. ------- ----_.. ._-- ---- - ---- - --- ----- ------ - ------8 .30 5.45 .915 .916 .915 2.2611,_~ ____._.55___ _.~_?_•.10 --- ..:......___•. 857 .• 857__~ __~ 657 _,- _2.11&-9 1.00 5.70 .958 .958 .958 2.364
-- ~O___ ______~ • 2 I} 5.85 .983 .983 .983 2.427_.__._-~- _•._-~-_ ...
2 1.50 5.60 .941 .941 .94.1 2.3233 3.00 5.25 1.051 1.051 1.051 2.593 .•..(1,.
.-----,----,- - _. .-._._- _._- - -----_._--,. 7.00 6.50 1.093 1.093 1.093 2.696
----------- --- --------,.---1---"---
------------------ ._--- ---------. f----------. ----------.-----------.-----
---------1----------- .-------------------------- .------
Table 4~l: (Continued)
----------------------48- -- ---.-------------~-------_._-----_._------- ----------
--- --------------_.,_.. _- - ----_._--------------------
------_._------R VOLUMETRIC CRITICAL MODIFIEDU SOLIDS DEPOSIT FROUDE
____________ _ N CONCENTRATION VELOCIT't NUMBER _ __. . ._t---4---------t-----o+__ (n-( I I )-U1 I) -<IV)-
---_._--_._----~..- .(PERCENT> . (FT/SEC)
1_--,- . 2._
345
------,----------_&-
._..-. • 05 :J • 7 0.10~.90
.25 4.50
.55 5.102.25 5.50
.. __~~ •.?Q ._ ." .....__? :'-0 _
.622
.655
.756
.857
.925_....•_.95.!!
.659
.695
.802
.909
.9601.016
.642 1.627
.576 1.715
.780 1.979
.884 2.2,.3
.954 2.418• 9 8 8 ~. 506
~_.__.._~-~.~"- -~-- - -~---_.-- .._~._--_ ... - -------- .__ ....... ~---- ~-'--,_..----_._-~..--..- ----'---1
.__:._'.~r~::. . . _.; :·_..:2_3
______________ . 4 _
.75__ ._ '1.90. 2.50
5.40
5.85 .8036.95 ._._ .9547.,.5 1.023
___L. 9_~_ ~ .. 1. 091
.803
.9541.0231.091
____. I Series_ BS-003 .{:~:~ig~-~-~-~r~~-~~ E~.~O •.~~~_~_M- .. _PIPE SLOPE = .027
--------_. "----.- ------------ ----_. ------ - --'- ------. -- -_._- ----- - .._------- ---'--'--' -_.._._--1 .75 6.15 .844 .621 .833 2.168
______~.. iL_ _.__~. 00 L! 1 0. ..__.. 975 _'__0 ••_~46 • 962._..__2.•.2.Q33 3.70 7.50 1.023 1.002 1.016 2.64"
____. .. . It 5.0_0 _ ..J_.75... .. .. 1. 06~ 1.035 1.050 2_!-7'_33_
-------------..--_=-se-r~e~-B-s~:::-{H:n~~~i~~~:::~0;~. i~. ~~~.--'---'-----~--.~-------~ -~.-~~.~~=--
- --.--.---_._" ._~._-_._ ... _. ----_.---. ._--_._- ---_ ..-.. . ----_._- _.,--~._-_._.. _.. -_._--~ .. -... _---- .. --_. ---_.__ ... ._----- -------1 1.30 3.40 .912 .972 .972 1.813
____________ .. _2. 1.9o . 3.85 1.101 1.101 1.101 2.05333~OO 4.~5 1.273 1.273 1.273 2.373
_~_____ _.~ _ 3.80 It_. 6 Q .1. 310_._J.• 316 _~_1._316_2_!_45.l_
---------'-............--.&._--------.&.-_--_......_------------- ._----Table 4.1: (Continued)
-- ~----- -·'·'-49----
- '-"- ._-------; --_.- -- --. - -~-...-.- -- .- -----_.. --_.
...
1.86 _
Solids Particle, ~o
if 0 if 00 if PP(0.88 nun) (0.45 nun) (3063 nun)
I\JloI
-o~' 0
1000· 0 1!Ir::I"-"'"""0_o_...-Lv-I1--~---,,-+----~t)-_·_~-t--,.-I.•;j. ~-..... o~ Ii. "~fo 0 e80 \J 0
~i 0 'rr======!:=====~0°.76 -
~~
- G - 10.60
G - 2• BS - 1
BS - 3o'
o:6.o'\l
oI . . • . I • • 0' , 1
0 1•0
:I
I.c[%]
I I0
1 . I I I
2.0 3.0 4.0 5.0 6.0
Fig. 4.1: Experimental Data from Lehigh Sand-Water and Plastic Pellet-Water Studies;Modified Froude Number versus Concentration~ Particle Diameter as Parameter
From a preltminary study, plotting Froude numbers (I), (II),
and (III) against solids concentra,tion, C, it was found that Froude
number (II) best correlates the data, including both horizontal and
sloping flow values. Further, Froude number (IV) plotted against
concentration, C, indicated no improvement in demonstrating the trend
. of results, and only increased the scatter of data. Lehigh values of
dID raised to the 1/6 power are very small and have little influence
on the correlation.
It is therefore that Froude numbers (I), (III), and (IV) are
no longer considered; the data are analyzed with Froude number (II)
and presented in Fig. 4.1.
Correlation of Data. A regression analysis was made to
*correlate 'modified Froude number (II) with the following parameters:
. concentration, C; concentration, C, and particle diameter, d; and con-
centrat ion, C, and relative particle size, diD. __
The regression functions take two forms: (1) A least squares
fit of modified Froude number, F , with concentration, C, written as:r
F- = k c~r ""1
(4.1)
where ~ and ka are evaluated from logarithmic values of the data over
five different particle size ranges, and (2) a least squares multiple
*The same was done for modified Froude numbers (I) and (III) and isgiven in Appendix C.
-51-
regression, using Gaussian iteration to fit modified Froude number,
F , to both concentration, C, and particle size, either as d, or ther-
dimensionless form, as din. These two regression functions are given
.-. .'~'~-: by.: . :.~ • __':'.#"n'\oo
k4 ~Fr =k C d
3
-~
,k' dint<;,F = C
r 3
(4.2a)
(4.2b)
The exponents, k4 and k , and coefficient, k are determined for the6 3
sand-water data and also for the total range of data, including plastic
pellet-water results.
An explanation of the multiple regression analysis arid a
statistical interpretation of the resulting equations are given in
Appendix c. It should be noted at this point that plastic pellets
data were eliminated from the analysis. The influence of 4 out of
50 data points is somewhat negligible and their imposition on the
general trend of results~ dictated by the 46 sand-water data points,
was felt to be of little value. The regression analyses reported in
Appendix C justify this reasoning.
TWo regression equations are found to fit the Lehigh data
particularly well: (1) Using only sand data, and assuming solids
concentration, C, to be the only important independent variable, the
best-fit equation is give~ as:
-52-
[1 - tan eJ (4.3)
The coefficient of correlation is 0.870. (2) Including the influence
of particle diameter, d, the following equation was developed for sand
alone:
[1 - tan eJ = 0.928 CO.106 do.o66 (4.4)
where the particle diameter, d, is in mIn. The coefficient of corre1a-
tion is 0.877.
Note that the value for exponent k = 0.106, given with. :3
Eq. (4.3),.is very close to exponent k4 = 0.105, given with Eq. (4.4).
Further, coefficient k3
= 0.928 in Eq. (4.4) differs only slightly from
coefficient ~ = 0.901 in Eq. (4.3). This similarity between the coef
ficients and exponents in Eqs. (4.3) and (4.4). is due to the almost
negligible effect of particle diameter, d. Equations (4.3) and (4.4)
are ~hown graphically in Fig. 4.2.
The regression analysis for the relation given by Eq. (4.2b)
is presented in Appendix C and shows that the relative particle size,
dID, has very little influence on improving the correlation given with
either Eq. (4.3) or Eq. (4.4).
It should be again noted that the form of the modified Froude
number, including a tan eargument, has been suggested to better corre-
late the Lehigh data. It is recommended that either Eq. (4.3) or
-53-
0.106F;. =0.901 C
t.
d=0.88 mm ". . 0.105 0.'056__d=0.45mmYfi.=0.928 C' d
v-;::::===c==~. [1 - tan e]
!2gD (ss-1)
0.601-:-- -1
•
0.76
1.001----=---o
1.a6
,VI~,
. .C[%]
·0 a.o 3.0 4.0 6.0 600
Fig. 4.2: Best-Fit Equations for Lehigh's Sand-Water Data Only; Modified Froude Number.versus Concentration, Particle Diameter as farameter
Eq. (4.4) be reliably applied in the design of sand-water transport
systems with galvanized or black steel pipes on a slope:
-0.10 < tan 8 <0.05. Either equation is certainly good within the
range of particle diameters tested at Lehigh: 0.45 < d < 0.88 nnn,
with Ii /d < 1.21.-So 60-
Relative Influence of Tested Parameters. Needless to say,
not all ranges of the parameters, D, d, ss' C, tan 8, d90/~0' and
e!D, have been completely investigated and never will be. However,
the resulting regression equations, Eqs. (4.1) and (4.2), offer in-
sight to the relative influence of some of the tested parameters on
the critical deposit velocity.
The influence of solids concentration, C, on the critical
deposit velocity is found in this study to be of primary significance,
particularly within a low-concentration rang~ of C < 7%. For con-
centrations above 5 to 10%, both Sinclair (1962) and Wilson (1965)
find that critical deposit velocities decrease with concentration.
A s~ilar observation was made in the present study when concentrations
exceeded 5%.;'
The particle diameter, d, has no direct effect on the crit-
ica1 aeposit velocity value within the range of particle diameters
tested in the present study, 0.45 < d < 0.88 nnn. However, with sus-
pensions of fine particles in the range d < 20 nnn, it is expected that
solids settling is sufficiently delayed to decrease the critical de-
posit velocity. This is reported by Worster et a1. (1955) and Gibert
(1960).
-55-
While the Lehigh data provide insufficient evidence that
relative density, ss.:!' expressed as (ss _1)°·5, is proportional to
the critical deposit velocity, other studies have made this verifi
cation. Sinclair (1962), howev~r, reports that (s _1)°';;' betters
correlates his data for iron-kerosene, sa~d-water, and coal-water
mixtures. Furthermore, Ellis et al. (1963b) conducted experiments
with nickel shots in water, finding that critical deposit velocities
were reduced for these solids of high density. They reasoned that
this was due to both the "elastic rebounding" of the particles, which
have large momentum as they strike the bottom of the pipe, and -the
increased lift forces Unposed by the liquid as the particles come
to a sudden rest at the boundaries. It appears reasonable to ques
tion the form (s _1)°·5 if it is used to determine critical deposits
velocities for solid-liquid mixtures other than sand-water. However,
for any suspension of quartz particles, (s _1)°·5 has been well foundeds
to best correlate the critical deposit velocity parameters.
The grain size distribution, ~o 1.250
' was also a parameter
felt to be unimportant in the present study. In addition, the Lehigh
sand samples were quite uniform and the effect of such a parameter could
not be tested. The problem of mixed sized samples is complicated in
that fine particles often create a supporting suspension for the
coarser particles. It is realistic, when designing for the trans-
port of a non-uniformly distributed material, to select an "effective"
mean particle size, slightly greater than d ,to account for the50 -
settling of the larger particles.
-56-
The relative material roughness, e/D, was assumed to be an
insignificant parameter in this study. Inclusion of this parameter
in the· correlation enters in the liquid head loss, and apparently does
not influence the movement of the solids phase. The present study
showed that for pipes of black steel and galvanized iron, material
roughness is of negligent concern in critical deposit velocity deter-...
mination. This is similar to what Durand (1953) observed with steel
and cast iron pipes. Only with very fine particles and pipe roughness
protrusions, which would disrupt the laminar boundary layer, might one
find it necessary to include the effect of e/D on critical deposit
velocity.
4.2 Comparison with Other Data
Particularly important in the present study is· the appli-
cabi1ity of the modified Froude number re1~tionship, given with
Eq. (2.13), for low-concentration mixtures, C < 7%. The strength
of the Lehigh data is in the range with 0.10 < C < 2.0%. The 10w-
concentration data are mainly responsible for the final form of the
modified Froude number relationship, as given with Eqs. (4.3) and
(4.4). In what. foltows we shall try to investigqte as to how other
experimental data compare with the present findings.
Sand-Water Mixtures. Many researchers have reported on
sand-water mixture studies, but from all of these, only the studies
n If .
by Gibert (1960), Fuhrboter (1961), Sinclair (1962), and Durand,
Smith, and Yotsurura, as reported by Wasp et a1. (1970), rendered
-57-
useful data for the present investigation. The ranges of parameters
investigated in these studies are listed in Table 4.2, and the data
are plotted in Fig. 4.3 for comparison with the Lehigh sand-water data
given with:
F = 0.901 CO. 10S
r (4.3)
Data were retrieved from only those studies which investigated
a "critical condition" identical to the critical deposit velocity, as
defined in the present study. However, it must be pointed out that a
certain amount of inaccuracy is inherent with any sediment transport
study and results will vary within the same testing system, let alone
from on~. system to another. In general, it is felt that the trend es-
tablished by Gibert's (1960) data, for d > 0.37 mm, is rather well re-
fleeted in the Lehigh sand-water data. It is recalled that Gibert
(1960) reports an exhaustive investigation obtaining 310 data points.
Of interest is also that the Sinca1ir (1962) and Durand (1953) data
are in reasonable agreement with the Lehigh findings'. Further, it is
noted that the Yotsurura data, reported by Wasp et a1. (1970), reflect
trends similar to the Gibert (1960) curve for fine particles.
Figure 4.3 together with the Lehigh sand-water data,repre-
sented with Eq. (4.3), suggest the following trends in the range where
c < 5%: (1) The critical deposit velocity, VC
' increases with solids
concentration, C; the increase becomes less evident as the concentra-
tion rises to 5%. (2) For particle sizes, d > 0.37 mm, the critical
deposit velocity remains practically unchanged with increase in d.
-58-
IVI\0I
Sediment Pipe Sediment Specific RemarksSize Size Cone. Gravity. --. --_.~. ._------_. ..-d [nun] D C Ps'P
_... . -.-....._......-._~ _.. -_.. . ~ .-- :.. ._.....- . _..._. ... ~.... _._.......- .~.. ~ -_. __... ........... ~.- -- .- - .,
(1952)* 0 0.44 2.65 Extensive rangeDurand • 2,.04 5.90 in. up to 15% 'sandI of 'parameters
water . tested
2.65 Vcobtained from
Smith (i955 )* 0 0.18 3.00 in. up to 26% sandI Vc vs. C plotwater
=:: 0.37 40.2 to 2.65 Best-fit curves-- on vc/Ji,D'vs.Gibert (1960)-0- 0.20 150.0 nun up to 15% sandI C
water plot
II II 0 0.27 2.64Fuhrboter (1961) • 0.53, 0.88 0.30 nun up to 25% sandI Vc
is reportedwater
(1961)* \l 0.232.65
Yotsurura~ 0.59, 1.15 4.25 in. up to 25% sandI Vc is reported
water
6 0.35 0.50, 0.75, 2.61 Vc obtained from.Sinclair (1962) A 0.68 1.00 in. up to 20% sand/ Vc vs. C plot
water
*Reported in Wasp et a1. (1970)'
Table 4.2: Range of Parameters of the Data Reported by Other Investigatorsfor SandlWater Mixtures; Data are Plotted in Fig. 4.3
•LehighSand-Water
Data
o· -0 0
- 0 -0-
0
."j
Gibert__ d>
0.37 rom-o-d = 0.20 rom
Durand 0Smith. 0
II "LEGEND Fuhrboter 0(see also Tab le 4.2) Yotsurura '1
Sinclair '8C [%]
3.0 . 4.0 6.0 6.0a.o1.0
vC [1 _ tan e]
. '2gD (s -1)'V s ~
o
1.00
1.26
*I 0.760'\
~0
*I
00.60
Fig. 4.3: Modified Froude Number versus Solids Concentration, Particle Diameteras Parameter (Data from Sand-Water Mixture Studies)
The Lehigh data exhibit this trend showing particularly good agreement
with the other data, and will give conservative desi~n values. (3) For
particle sizes smaller than d = 0.37 mm, the critical deposit velocity,
Vc' decreases with decreasing d. It is expected that this decrease in
Vc levels off for very fine particles, but the data reported give in
conclusive verification of this.
Neither particle size distribution nor the pipe material
roughness were considered to be of importance in this comparison.
Solid-Liquid Mixtures other than Sand-Water. To show the
general usefulness of the modified Froude number, data from other
solids-liquid mixtures were studied. Wasp et al. (1970) report data
from Wicks and Moye on the investigation of sand-kerosene and sand-
oil mixtures, Sinclair (1962) reported on iron-kerosene mixtures;
and Wilson (1965) on nylon-water mixtures. Again, the data are com-
pared with the Lehigh sand-water data, as shown in Fig. 4.4; the
ranges of parameters are listed in RabIe 4.3.
Whether the density parameter, given as (s _1)° 0
6, bests
correlates solid-liquid mixtures other than sand-water is difficult
to assess from the reported data. Higher relative density mixtures
tend to decrease the critical deposit velocity value as demonstrated
by the Sinclair (1962) and Wasp et al. (1970) data, and~s explained
in Section 4.1, after Ellis (1963b). Whereas, the lower density sus-
pensions reported by Wilson (1965), and shown with the present study,
fall significantly above the Lehigh sand-water data.
-61-
,0'1NI
Sediment Pipe Sediment Specific RemarksSize Size Cone. Gravity-
d [nun] D C p/pI
Sinclair (1962) 0 0.12 0.50, 0.75, up to 20%10.37 Iron/ Vc is reported on
• 0.09 1.00 in. 0.78 Kerosene Vc vs. C plots
Wilson (1965) 0 3.88 3.48 in. , up to 20%1.14 Nylon/ VC is shown on
3.69 in. Water head loss curves
6 2.65 SandI .Wicks and * 0.25 1.05 in. , 1:0% 0.91 Oil Vc is reportedMoye (1968) 5.50 in.
A 2.65 SandI0.81 Kerosene
Lehigh (1971) ~ 3.63 6.00 in. up to 5%1.38 P1astic/ Vc is reported
Water
*Reported by Wasp et a1. (1970)
Table 4.3: Range of Parameters of the Data Reported by Other Investigatorsfor Solid/Liquid Mixtures other than Sand/Water; Data Plottedin Fig. 4.4
vC [1 e]- tanV2gD (ss -1)'
1.26
LehighSand-Water
r--------IGiI-=::!::;;;;;;;;_--~D~a~t~a~====~~::;::t=:::=====~===::~+~-11.00 r
8\ '.' 0 •I 0.76
0'10UJ
I 0~
~LEGEND (See also Table 4.3)
ASinclair (Iron/Kerosene) 0
0.60Wilson (Nylon/Water) 0
8 Wicks and (Sand/Oil) 8Moye (Sand/Kerosene) V;
!'j Lehigh (Plastic Pellets/Water) ~C ['Yo]
0 1.0 2.0 . 3.0 4.0 ' 6.0 8.0
Fig. 4.4: Modified Froude Number versus Solids Concentration, Particle Diameteras Parameter (Data from Studies of other than Sand-Water Mixtures)...
// Although these res~lts are inconclusive, it is suggested
to use the modified Froude number relationship, in the form given
with Eq. (2.13), till further data on non-sand-water mixtures become
available.
4.3 Engineering Application
An engineer, confronted with the task of designing a solids
transport system, finds that a theoretical application of critical
condition transport has many limitations. In another instance, he
. may be unable to apply one particular approach, because its validity
has not yet been tested for the type of mixture slurry he is con
sidering. Furthermore, he is usually provided with little or no
information on the economic factors to be considered in installation.
and operation of the system. The basic problem in design is one of
safe operation and minimization of the costs to transport the mixture.
The critical deposit velocity relationship, as defined in
the present study with either Eq. (4.3) or Eq. (4.4), provides the
designer with a useful tool with which he may define the optimal
operating conditions of the system. To ensure safe, uninterrupted
transport of the mixture, the designer must also properly select
pump, pipe material and instrumentation, after consideration of basic
hydraulic parameters and· power requirements. Condolios (1963b & c)
and Graf (1971) treat the subject of solids pipeline operation with
considerable proficiency.
-64-
/ 4.3.1 Economics of Solid-Liquid Transport Systems
A rather attractive feature of the solid-liquid transport
pipeline is the minimal cost required for operation and maintenance,
as compared with the conventional means of transporting solids. In
'addition to the revealing economic advantages, pipelines are ammenab1e
to automation, are dependable, and can overcame both natural and man-
made- obstacles..
Operating costs are minimized when the power required for
transport is held to a minimum, however, certain precautions mUst
be taken. The minimum power input and the minimum mixture head loss,
i • are coincident and identify a region in which the system may bem
come unstable. This leads inevitably to plugging of the system.
Operation in this region is unsafe, and slightly higher flow veloc-
ities should be maintained to avoid system instability. Condolios
et at. (1963b), Ellis et al. (1963a), and Wilson (1965) discuss ap-
plication of the minimum power requirement in design.
The critical deposit velocity, VC
' is often found within
the region of ·instabi1ity. It has been observed by Cond91ios et a1.
(l963b), Wilson (1965), and within the present study that the re-
lationship between critical deposit velocity and the velocity cor-
responding' to the minimum head ioss is as given with Fig. 4.5. Vcis higher than the velocity associated with the minimum head loss at
low concentrations - however, the opposite is true for C > 5%. An
explanation for this occurrence is reported by Wilson (1965). The
heavy line in Fig. 4.5 represents a reconnnended envelope for deter-
mining the stab Ie operating flow ve 10city.
-65-
1.0
0.1
0.01
i (log)m
MixtureHead Loss
20/0
MinimumHead LossLine
i (log)mMixtureVelocity
10Fig. 4.5: Critical Velocity and the Velocity Corresponding
to the Minimum Head Loss
Condolios et al. (1963c) report on instability of the pump
characteristic curve, due to the fluctuations of solids concentration
during operation. The designer must consider the characteristic stage-
discharge curves of the pump in comparison with the mixture head loss
curVes for the pipeflow to ensure stable design. '
A method for optimizing solids concentration, C, and pipe
size, D, was reported by Hunt et al. (1968). Although some preliminary
economic considerations of solids pipe lining have been reported by Wasp
et al. (1967), the relationship between hydraulic and economic decision
variables had not been presented analytically. Hunt et ale (1968)
-66-
/
/ minimize a function containing seven cost groups and hydraulic parame-
ters,with respect· to C and D. The response surface generated by this
cost function yields various combinations of C and D and the most suit-
able are selected for design;.
The engineer, in designing a solid-liquid transport system,
must concern himself with some basic considerations:
Installation:
(1) .Physica1 characteristics of the mixture(2) Adequate pumping facility(3) Flushing and drainage(4) Pipeline wear and corrosion
Operation:
(1) Physical characteristics of the mixture(2) Stability of pipe flow(3) Stability of pump operation(4) Optimum delivery of solids
Lowenstein (1959), Ellis et a1. (1963a), and Roberts (1967)
present different methods for designing economically practical trans-
port systems. Use of the Lehigh findings as a basic criterion in the
design procedure is presented now.
4.3.2 Application of the Lehigh Findings to Design
The "critical condition" has seldom been used as a criterion
for designing economic transport systems. The apparent reason is that
relationships for the critical deposit velocity have been vague in
conclusive evidence and thus, engineers have retained little con-
fidence in their application. The Lehigh findings provide the de-
signer with that criterion which will minimize the cost of operation
and ensure safe, uninterrupted flow conditions.
-67';
For designing a system to transport sand with particle
diameters, 0.45 < d < 0.88 nun, in water, Eq. (4.3) is reconunended,
and is rewritten here as:
(4.3 I)
...
If the sand particle sizes are larger, d > 0.88 nun, Eq. (4.4) is
reconnnended and can be rewritten as:
(4.4' )
Equation (4.4') will give more conservative values for Vc than Equa~
tion (4.3'), as particle size, d, increases in size over 0.88 rinn.
For particle sizes smaller than 0.45 nnn, neither Eq. (4.3') nor
Eq. (4.4') are reconnnended. One is then referred to Gibert (1960).
Roberts (1967) presents a general method for extrapolating data to
regions outside o·f the tested bounds, app1:Lcation of which would
enable more extensive use of the Lehigh equations.
To illustrate general application of the Lehigh critical
deposit velocity equations, Eqs. (4.3') and (4.4'0), and Fig. 4.2,
two typical design problems are examined.
Example (1). Suppose a long distance miner~ls-watermix-
ture transport system is to be designed for a certain delivery rate
of "solids,Q . (defined as tons/mile/hr), and given with diameter, d,s
-68-
,I
I and specific gravity,
to minimize costs?
s •s
/
What parameters must the designer consider
Delivery rate, Q , is' defined with the following relations
ship:
Qs =~ c = V A C (4.5)m
I
where ~ is t~e mixture flowrate. It is recommended that the critical
deposit velocity criterion, resulting from the present study, be em-
ployed. Equation (4.5) is therefore considered to be minimized with
respect to unit costs by replacing Vm with Vc and rearranging:
Q' = IT V c if5 4 c
where Q' now represents optimum solids throughput.s
(4.6)
If particle diameter, d, as an example, is slightly larger
than the range of particle sizes tested in this study; i.e.,
d - 0.10 mm, we can substitute Eq. (4.4) into Eq. (4.6) and obtain:
rearranging:
(4.7)
Q' = 5.85 cLll do.o6 if·S (5 _l)OoS55.
-69~
-1(1 - tan e) (4.8)
Note that this equation is similar in form to the relationship given
by Eq. (2.10), but it is pointed out that the exponents and coefficient
of Eq. (4.8) are constant over the entire range of Lehigh data, and the
relation can be extrapolated in many instances to include parameters
outside these tested ranges •.
The pipe slope, tan e, is identified, through a topographic
survey, as to where it will be a maximum. From Eq. (4.8) the most
equitable combination of concentration, C, and pipe size, D, can be
determined throu8h trial and error. If concentration is larger .than
5%, extrapolation of the Lehigh data must be undertaken with caution.
If the particle diameter, d, of the slurry to be transported is
0.45 < d < 0.88 mm, Fig. 4.2 can be used directly and optimum modified
Froude numbers located readily.
Example (2). Consider the design of a pressurized solid
waste disposal system. A difficulty encountered with the hydraulic
. transport of solid wastes is the identification of slurry character
istics. Non-Newtonian suspensions cause a problem which is not con
sidered within the scale of this study, however, real concern is for
the settling and possible clogging due to grit and sand in the mixture
slurry.
If a system is designed to handle a specified concentration
of settleable solids from domestic disposal units, will the 'working'
operating velocity become a critical deposit velocity, or more
seriously, a sub-critical, unstable flow velocity, if solids concen
tration is suddenly increased? The characteristics of the grit
-70-
i. I
concentration, given with d and (s -1), dictate which Lehigh designs
equation is to be used. From either Eqs. (4.3), (4.4), or Fig. 4~2,
the variation in modified Froude number, with increase in concen-
tration, C, is observed. Subsequently, a new value for Vc is. defined
and compared to the original conservative operating velocity.
Th~ application of the Lehigh equations can be extensive,
considering that extrapolation is performed with caution, and one
understands clearly the definition and relative influence of each
parameter.
-71-
5. CONCLUSIONS
The critical deposit velocity, VC
' tested in the form of a
modified Froude number, is correlated with other parameters, which is
significant in the solid-liquid transport problem, over the following
ranges:
0.01 ~ C ~ 7.00 %0.45 ~ d ~ 0.88 rom4.00 < D < 6.00 in.- -
-0.060 < tan e < 0.0271.07 ~-~o /d5c~ L.21 .
0.00009 ~ e/D ~ 0.00032
From a dfmensional analysis of these parameters, a modified
Froude number relationship is developed, as given with Eq. (2.13).
The relationship is tested for sand-water and plastic pellets-water
transport. Data from the sand-water tests ..exhibit the following:
(1) Agreement with the Gibert (1960) curves for
particle diameters, d ~ 0.37 rom.
(2) The increase in· critical deposit velocity, VC
' be
comes less evident as solids concentration, C, rises
to 5%; above 5%, Vc tends either to remain constant
or decrease with increase in C. [This was also ob-
served by Sinclair (1962) and Wilson (1965)J.·
(3) For particle ~izes, d ~ 0.37 rom, the critical deposit
velocity remains practically unchanged with increase
in d. ,.
(4) The critical deposit velocity is higher than the
-72-
velocity associated with the minimum head loss at low
concentrations; however, the opposite is true for C> 5%.
Findings from the plastic pellet-water test data were incon-
clusive.
A regression analysis, made to correlate the Lehigh data,
'"
shows that the modified Froude number is highly dependent on concen-
tration, C, slightly affected by particle diameters, d ~ 0.37 mm, and
hardly influenced by relative particle size, diD. The regression equa-
tions which best fit the data and are in reasonable agreement with data
from other sand-water studies, are given with:
"
v'C [1 - tan eJ = 0.901 CO,lOS (4.3)
- tan eJ (4.4)
Although the reliable application of these equations for
solid-liquid mixtures other than sand-"water',has been .inconclusively
resolved, it is suggested to use Eqs. (4.3).and (4.4) in their present
form till further data on non-sand-water mixtures become available.
lbe Lehigh critical deposit velocity equations give con-
servative values, and are presently the only relations available for
predicting critical deposit velocities for low-concentration solid-
liquid mixtures. It is recommended that either Eq. (4.3) or Eq.(4.4)
be 'used as a critical deposit velocity design criterion, certainly with-
in the range of parameters tested in the present study, and 'cautiously
in ranges of parameters extending outside of the tested bounds.
-73-
//
APPENDIx A: EVALUATION OF LOOP READINGS FROM PROGRAMMED OUTPUT
Determination of ~ and C
The "Loop System" became a useful tool for quickly determining themixture flowrate, ~, and 'solids concentration, C,once the programmedoutput was plotted. Enlarged sections of Chart 1 and Chart 2, fromFig. 3.5, are shown in Figs. A.l and A.2, respectively. With referenceto these two charts, the determination of ~ and C from loop head lossreadings will be examined-
System water temperat~res during a test run sometimes increasedfrom 60°F, at the beginning of the run, to 100°F, after high flowratetesting bf a large solids concentration mixture. The loop indicationof mixture flowrate is appreciably affected by temperature changes, andsince it could not be easily controlled, readings at temperatures ofboth 70°F and 90°F were plotted on Chart 1. Water temperatures wererecorded during the progress of a test and employed in the evaluationof ~ and C, but they are not reported in the data of Appendix B.
Recording for one test, lihR, the riser pressure drop, and,·lihD,the downcomer pressure drop, the concentration, C, would normally bedetermined immediately from locating (lihR-lihD) on Chart 2, since .thisrelationship is hardly a function of flowrate,~. Proceeding then toChart 1 and knowing C, (lihR+lihD), and temperature, Qm' would be located.
However, through repeated' clear-water calibration of the loop system, riser readings were observed to be consistently greater than thoseof the dowilcomer and generally increasing with mixture flowrate. Thesedifferences were attributed to insufficient transition length, incompletely dissipating the local turbulence effects following the elbow'bends. The trend of deviation is shown in' the "correction curve""belowChart 2 in Figs. 3.5 and A.2*. The difference was assumed to be equally shared by the two vertical sections, such that the (lihR+lihD) readingneeded no correction. The (lihR-lihD) reading acquired the full correction directly. To better illustrate the additional implications andconvergence on Qm andC values, an example is presented.
In Series G-02-3 of Appendix B (tests of coarse sand transportthrough a downward sloping, 4-inch galvanized pipe), the first set ofloop readings recorded are:
* .For later investigations of plastic pellet and additional low concen-tration sand flows, the transition length before the loop pressure tapswas extended 3 ft. This greatly reduced the correction curve toanearly constant - 0.2 values over the entire range of flowrates.
-74-
/
6hR
= 33.00 in.
6hn = 11.05 in.
Consequently, resulting in:
6hR
- 6hn = 21.95 in..
The system temperature for this particular run was recorded at 82°F.
A first approximation of concentration, C, obtained from Chart 2,would be 10%. On Chart 1, Fig. A.l an 80°F recording for 10% mixtureconcentration would fall at point ~ in correspondence to the summedhead loss value at @. Interpolated to an 82°F reading, point ®shifts to @, locating Qm = 410 gpm. In Fig. A.2, the correctionvalue at @, corresponding to Qm = 410 gpm, is -1.35 in. Applied tothe head loss differential at point @ on Chart 2, an adjusted differential head loss, of 21. 95-1.35 = 20.60 in., is located at ®. Theresulting C = 10.5% was considered close enough to the original assumption of C = 10% to warrant acceptance of the values:
~ = 410 gpm
C = 10.5%
Further iteration of this procedure was seldom required, if anapproximate correction value was considered in the first attempt.
When both the flowrates, Qm' and volumetric concentrations ofsolids, C, were in their upper ranges, discrepancy of loop.readingsfrom Prandtl and Pitot tube observations was often detected. Adjustment of these readings is now discussed.
Adjustment of ~ and C in the HeterogeneousFlow Regime
It was observed that the magnetic flowmeter readings were systematically higher than the velocity readings given by the loop.Further, visual observation of the flowing mixture indicated an apparently greater volumetric concentration of solids than determinedby the loop. These discrepancies were particularly noticeable at
-75-
-a::W 26....~li-0.
24z-c
.c<J
. I-..J I 220\I a:
.c<J
20
-Q za::
.c OW +2<J - ....
.... <tI u3=
Wa: O::li-
.c 0::0<J 0u .
°z-
. 1
.~
'1) C=14°/o!-
~
C=13°/o
- II ~- .
;
--- c= 12°k
-
® C=IIO/o
-C= 10.5°/0 1.0 (cfs)® 10~8 1°·9 I
I I I350 400 450 (gpm)
f- .-@
---350 400 450 (gpm)
I I I .
Fig. A.2: Enlargement of Chart 2 and Correction Curve in Fig. 3.5
..
(cfs)
(gpm)450350 400 Qm=410gpmFig. A.l: Enlargement of Chart 1 in Fig. 3.5
0.7
52
®
- 50 r0:::
/ ~w -~ II« (J.3=
/.u- 48...... 0...... --70° F /• z- gO°F
Q /.c46<l /+
a:
/. .c<l @
44
.,
flowrates and concentrations above the critical condition, well intothe heterogeneous flow regime.
To assure confidence in the "Loop System" recordings of mixturefloWrate, Prandtl tube traverses for clear-water flow were run over a.range of flowrates between 160 and 600 gpm. Reliability was placed inthe Prandtl tube results and were used to calibrate the Foxboro Magnetic Flowmeter. Within the range of flowrates tested, the flowmeterwas found to be consistently indicating flowrates 12.5% in excess ofthe actual flow conditions. It was felt that the magnetic flux methodof determining flowratewould be accurate in measuring mixture flowupon the entrainment of solids in the system, such that loop readingscould be evaluated from flowmeter recordings using the 12.5% correction.Flowmeter indic~tions of Qm were indeed found to be greater than theloop, and the discrepancy increased with larger flowrates and largerconcentrations, although never exceeding 8%.
A Pitot tube sediment-sampling device was employed to evaluateloop indications of solids cbncentration. The copper sampler wasunable to withstand the sand-blast effect of the larger particles,however, samples were obtained for the finer sand. The difficulty ofvelocity flow equalization within the system and sampler was apparent,but an insignificant deterent for establishing some degree of reli-
.~ ... ability in the sampling results. It was discovered that the· concentrations evaluated using the sediment-sampling device were also largerthan those given by the loop. The discrepancy increased with flowrateand solids concentration to magnitudes of up to 50%.
Explanation of these unexpected discrepancies implicates a studyin itself, and within the scope of this study, only a method of adjustment can be determined. The method recommended for adjusting theheterogeneous flow regime data is explained in what follows.
Considering the same set of data just examined, a flowmeter reading and Pitot tube sample might have respectively indicated:
QF = 490 gpm (actual QF
=> 490 x 0.89 = 435)
*C = 14%p
Digression from the loop readings is markedly significant and is represented as:
*The sediment-sampling device was clogged and damaged when testing thecoarser sand so that the method of correction used for fine sand couldonly be assumed applicable to the coarser sand concentrations.
-78-
/ QF - Q 435 - 410 = 25 gpm (6% discrepancy),I =m
I C - C = 14 - 10.5 = 3~5% (33% discrepancy) .p
The sum of the "Riser" and "Dowricomer" head readingsby locating on Chart ~ as illustrated in Fig. A.l,a corrected value at h:
(MlR
+ llhD
) ~ 51.0"corr
was first adjustedpoint@ indicating
!i.j
(
The deviation between flowmeter and loop reading is denoted as:
(llhR
+ llhD
) - (llhR
+ llhD) ~ 7.0"corr
It is then observed that the identical adjustment of head differencemost completely corrects the concentration reading. This is shown onChart 2, of Fig. A.2, whereC of 14% is located at Q), following theappropriate adjustment of both (llhR~llhD) and Qm'
These findings were consistent at all concentration and flowratecombinations and became an integral part of a venturimeter investigation, Robinson et al. (1970). It was noted that at low flowratesand low concentrations, both the magnitude of deviation and percentagecorrection were no longer s~gnificant to warrant serious concern. Sincethe primary interest in the present study was in the critical velocityrange for low concentrations, the minor adjustment, as discussed in thissection, was deemed unnecessary. However, when applying heterogeneousflow data, from Appendix B, there should be consideration of appropriateadjustments, as just illustrated. .
Figure A.3 is a useful tool for approximating the necessary corrections for any combination of Qm and C up to 600 gpm and 15%, respectively. QI and CI represent the recommended percentage increaseover the loop values.
-79-
•
~.. .,.......'. .~ .... .-. -,
.. " "-." .. ".. " ,'":. .."
Q -6°/":::.::'.:.1- 10 ' •• ".
C -35°1 •• ::,:-: '.'I-1o " ....:.
.. ...... .. .. .... ,,-...... .. ','
.. ..".. " ...... " ..'I '.-;, ..::
~ .. ':'.. '".... ".... ..": .. ""', ....:.............. '-
..," .,' .....• ",' .............. .. . .." ...... ', " ..,.... '....
Q -4°1 :,'. ":"1- 10 ..:.' ':"
C =20% "::':':-:::"1 ",,: ...,,, ..".' ...- ,. :: ....
.. -: ..'::. "' ..
":~.:.. .. ":": .. ,.. ".. .. ...... :: '.' :.':.......... ~.. .. •••;-.n.~·_-.::.:J....... ''''::::.
... : .. ,',.. -'. ...
' ... . .. ". .'.... "" ..... .. :.' ., .'
"" ..'.,'., "
,,- .: ..........: .. '.:..... ... ,',
" .... ' .'.: .
'. ,'.-. ',=;..." .......... ..: .." .. ".~... .. .."" ...... ..
Q 1 =2 o/~'::~-;.:':'"''C
1=5°k "::;"}:,.:... ' .... ... . .......' '. ".. ' .......: ' ..': . ' ....':.',,'.'. " :. ':'., ...... .. .. :.:: : .', :.', " ,,:: "
200
400
Eo
ECO'--
600~.r:-::;.~.--------:-.":T':r-:-------"':"""'I'""::------Q-=-a-0/c-o':":"'.:·:'(":'"".::.....-,::::.
'.' .:: ': . C:= 50 o/~··::~~
o 5 10 15
Fig. A.3: Pe~centage Increase Corrections of Both Flowrateand Concentration for all Combinations of theTwo Parameters
-80-
APPENDIX B: TEST DATA COMPIIATION
Parameters of pr~ary significance in their effect on the criticaldeposit velocity are: The inside pipe diameter, D, the pipe materialroughness, e, the slope of the pipe, S, the mean sed~ent particle size,~o' with consideration of the non-uniformity coefficient, dgo/dso' andthe specific weight of the solids, s. These parameters have been varied to determine how each enters intg the modified Froude number re-lationship, defined in the text of this paper. The series of tests arecoded with the following convention:
Horizontal- Downward, -0.060- Upward, +0.027
o - ~o= 0.88 mm , ~Q = 1.21 , s =2.65·"'6Q S
00 - dso= 0.45 mm , ~~ = 1.07 , ss = 2.65
PP - e1s0= 3.63 mm , s = 1.38s
SOLIDPARTICIES
4 in. ¢ Galvanized, € = 0.00001 ft •. 6 in. ¢ Black Steel, € = 0.00016 ft.
RUN NUMBER(Indicates change in con
centration
{~l
~---------PIPE SLOPE
---PIPE {B~
G - 01 - No.
G - 02 - No.
G - 001 - No.
G - 002 - No.
BS - 01 - No.
BS - 03 - No.
BS - 001 - No.
BS - 003 - No.
BS - Ppl - No.
t
-81-
ITest Section:
Explanation of the Table Headings
(Over a ~t = 3.60 m (= 141.8 in.) test section the headloss was determined; U-tube manometers were used).
or
Measured mixture head loss (in inches of a liquid with aspecific gravity pf Ss = 2.95).
(in inches of water)
Mixture head loss gradient (calculated from ~~95).
Loop Readings: (The ·'Loop System" developed by Einstein and Graf(1966) was used to simultaneously determine the mixture f10wrate, ~, and the solid phase concentration,C.
Head losses in the Riser and Downcomer sections (3~inch
pipe, 1.50 m (=59.1 in.) long; U-tube manometers areused).
Vm
~-MtD
&R-MlD(cori.)
C
Comments:
Sum of the head losses •
. Mixture f10wrate, according to theory of Einstein and Graf(1966), from the sum of the head losses.
Mixture velocity in test section determined with continuityrelation.
Difference of the head losses.
Correction of above from predetermined clear-water testcorrection curve.
Concentration, determined according to theory of Einsteinand Graf (1966), from the difference of the head losses.
Commentary of observations in Plexiglas section on the conditions of sediment transport and deposit.
Each table is summarized indicating the critical condition; this1s the critical velocity, VC' for.a specific concentration, C.
-82-
/
Some Remarks to the Figures
Plotting of the data follows on mixture head loss versus mixturevelocity graphs. These graphs show the variation of critical velocity,Ve, with a change ,in solids concentration. Constant concentrationl1nes are fitted to the data, and the critical velocity for a particular concentration, subjectively observed as the velocity at which anon-moving bed forms on the bottom of the pipe, is located. At velo-cities below the critical, equi-concentration (constant "moving" concentration) lines are dashed (---), while the diminishing concentrationline for a particular run, is drawn solid (--).
The relationship between critical velocity and the minimum headloss condition can' be qualitatively examined.
Some Remarks to the Data
It was explained in Appendix A that some of the data recorded athigh flowratesand high solids concentrations require adjustment according to observed Prandtl and Pitot tube corrections, as shown inFig. A.3. These adjustments were found to be insignificant in thecritical velocity ranges~ hence, the data remain as recorded from theloop readings.
It is also to be noted that some drafting errata in pipe roughnessvalues, e, have been corrected since the first reporting of this data,Graf et al. (1970). Except for the inclusion of test data from plasticpellet and additional low concentration sand mixture flows, the originaldata remains una1tered~
, These more recently obtained data were not included on the headloss figures, but are of extreme significance in the final evaluationof this problem. They are tabled under Series. G-Ol-6 to G-01-1l,G-OOl-S to G-OOl-IO, and BS-PP1-1 to BS-PPl-4 inclusively. It has beennoted in the text that for these studies, an improved clear-water cor- 'rection value was applied •
.,
-83-
......~ .
" .,- ~'-. ~---""." "-
15
...."': ..
0= 4 in.dso =0.88mm
S= -0.060E =0.00003 ft.
/10%10.5 5 0
/0
/_10.0 .6.59.5: .4.75
~_.9.0
8.8.-8.5
3 45678910
Vm' Ups), MIXTURE VELOCITY
0.3
0.2
0.02
. 0.01 L...------L.__-1-_.....1-_.l---'---I.--l--l .....1-
2
en 0.1en 0.090-J 0.080<t
0.07LLI:J:
LLI0::::::> 0.05~
'-Clear Fluid~~ 0.04
Ei i
..cl~ 0.03<1<1, ,
/
Plot of Series C-OI Data
-84-.~
,' ..
1
di1i_tf..!st sect on_ oop rea ngs
6,1- ( 6h )Al~ ~hD AhR+l\~ ~1 V 6h
R-Mln ~IR-hhD C COm'1ENTS'"1.95 ~,f;. fA
m corre.c ted[in.) [i.n. ] [In. ] [in.J [gpm] [fps] [i.n. ) r' '1 ['1oJ.i.. n . ~~
- - ......... -- _..
10.50 0.158 29.55 18.90 48.45 440 11.15 10.65 9.05 4.75 Everything moving
8.30 0.114 23.10 14.65 37.75 385 9.7 8.45 7.35 3.75 Suspended and, bed load
6.40 0.088 17.65 1l.0 28.65 335 8.45 6.65 5.75 3.0 Su spended andbed load
5.60 0.077 1l.60 7.70 19.30 275 7.0 3.90 3.30 1. 75 Moving bed,3.30 0.0455 8.10 5.90 14.0 230 5.85 2.20 1. 70 1.0 Pulsating, sliding
bed2.80' 0.039 6.40 4.80 11.2 205 5.2 1.60 1.20 0.50 Pulsating, sliding
bed
2.30 0.032 5.30 4.20 9.5 195 4.95 1.10 0.70 0.50 Slowly moving bed
ICRITICAL l2.30
.0.032 5.30 4.15 9.45 195 4.95 1.15 0.65 0.50 Just below critics
I00V1I
Series G-Ol-l
. . {c = 0.50%. CRITICAL COND ITION .
V = 5 0 fpsC .
1 d'.... teEit scction __... 001' rea LllgS _ -,th1. 95
(~) l~hR i\hJ) AhR+Ahn 0 V ahR-t.hD6hR-Ahn C COMHENTSat '11\ m,m
corrected(in.) [in.) [in.] [in.J [gpmJ [fps) [in.] [in.) [%]
6.40 0.088 19.40 10.60 30.0 350 8.9 8.80 7.80 4.0 Suspended andbed load
5.20 0.0715 13.80 7.70 21.50 290 7.35 6.10 5.30 3.0 Everything moving4.80 0.066 11.80 6,.80 18.60 270 6.85 5.00 4.40 2.25 " "4.30 0.0592 10.40 6.40 16.80 250 6.35 4.00 3.50 l.8 Moving slowly3.90 0.0535 8.80 5.50 14.30 230 5.85 3.30 2.80 1.50 Moving bed,
thickening layer3.30 . 0.0455 7.10 4.75 11.85 210 5.35 2.35 2.00 1.0 { neposi t· bed
CRIT;E,!"fl
3.50 0.0481 7.60 4.95 12.55 220 5.65 2.65 2.30 1.3 Bot tom limit ofmoving bed
2.60 0.0358 6.0 3.90 9.90 190 4.3 2.10 1. 70 0.8 Below cd t iea1
Iex>0\I
Series G-01-2
'. CRITICAL CONDITION {~Vc
= 1.00%
= 5.5 fps
- . I
/
di1 Pt f;.. e6;' ,sec Lon .... ! 00 rea ngs
t.hl . 95( ,Ah,) AhR AbD AhR+AhD Qm V' AhR-tlhD AbR-AhD C COMMENTS
tit m ~corrected
[tn.J (in.] [in.) (in. ] [gpnl] [fps] [in.) (in.) [~~J
9.30 0.128 23.95 11.10 35.05 . 375 9.55 12.85 11. 75 6.0 Everything moving
8.10 0.111 20.35 9.0 29.35 340 8.65 ' 11. 35 10.35 5.3 " "6.80 0.094 16.60 7.50 24.10 310 7.9 9.10 8.30 4.25 " "5.80 0.080 12.00 ' , 5.75 17.75 265 6.7 6.25 5.65 3.0 Sliding bed
5,.00 0.069 10.30 5.10 15.40 235 6.1 5.20 4.70 2.5 Pulsating bed
4.50 0.062 8.90 4.75 13.65 230 5.95 4.15 3.75 2.0 ~ed slowly moving
ICRITICAL I4.20 0.058 8.25 4.50
.12.75 220 5.6 3.75 3.35 1. 75 Non-moving bed
2.60 0.036 4.50 2.95 7.45 160 4.1 1.55 ' 1.25 0.6 Flatbed
1. 70 0.0235 2.90 1. 50 4.40 125 3.2 1.40 L10 0.5 Long dunes
2nd Run
2.20 0.168 31.85 17.70 49.55 450 11.45 14.15 12.55 6.5 Everything moving
9.50 0.131 24.55 12.50 37.05 390 9.8 12.05 10.85 5.5 " ' "~
6.70 0.092 16.35 8.35 24.70 310 7.9 8.00 7.25 3.75 " "4.80 0.066 10.15 5.30 15.45 240 6.05 4.85 4.35 2.25 Sliding, pulsating
bedI'
4.20 0.058 8.75 5.10 13.85 230 5.85 3.65 3.15 1.6 Bed slowly moving
4.10 0.0565 8.55 5.75 .13.5 225 5.75 3.60 3.10 1.6 Critical
3.90 0.0535 7.70 5.95 12.45 220 5.65 2.95 2.65 1.5 Deposit
3.20 0.044 3.75 5.60 6.60 145 3.7 . 0.t9O 0.60 0.3 Flat bed
•00"'-J
•
CRITICAL CONDITION: C = 1.75%, Vc = 5.75 fps Series G-01-3
.... '.\
. .- ~, ... : ~ .~. "
... ," ,>~~'. -.' .(
e
loop rcaJingste t section.. 8 - - -_._ .._._._.. _.~-'"
bh1. 95 (...ML) 6hR ~hD ~lhR+~hD ~1 V 6h
R-6hn AhR-Ahn c Ca1HENl'S
~-L' mm
-~ correc ted..
..
[In.J (in.] [in.] [In.] [gpm] [fps] [in.] [in.] [%] ; /.;. ~
~< ," .10.20 0.140. 26.75 12.10 38.85 390 9.9 14.65 13.45 7.0 Suspended and
bed load ..
7.70 0.106 20.10 . 8.70 28.80 330 8.35 11.40 10.50 5.5 Suspension, mostlybed load
5.90 0.081 11.85 5.50 17.35 260 6.6 6.35 5.75 3.0 Fast moving bed
4.70 0.0645 9.25 4.65 13.90 230 5.85 4.60 :4.10 2.2 Sliding bed
4.40 0.0605 8.55 4.50 13.15 225 5.75 4.05 3.55 1.9 Just above Vc4.20 0.0578 7.95 4.20 12.15 220 5.65 3.75 3.25 1.8 Non-moving bed
(CRITICAL],3.50 0.0481 6.65 3.85 . 10.50 190 4.8 2.80 2.45 1.2· Flat bed
2.20 0.0302 3.60 2.10 5.70 140 3.6 1.50 1.20 0.8 Flat bed--thinning(long dunes)
1.50 0.0206 2.40 1.50 3.90 120 3.1 0.90 0.60 0.5 6' long dunesat 2 intervals
4.40 0.0605 8.50 4.0 13.50 230 5.85 4.50 4.0 2.0 Scour (long impu1svariations)
,(Xl
·00,
Series G-01-4
{
c = 2.00%CRITICAL C0NDITION
VC' = 5. 75 fp s
loop e-d.lngtit t,-,CS acc on_ ~. r Q .4. B
lIh1 . 95(...ML) AhR lIhD 6ha+AhD Qm V ~hR-6hD 6h
R-t:.h
DC C0Mt.1ENTS. At-. mm corrected
[in.) [in.] [in.J [in.J [gpm] [fps] [in.] [in.] [%).. - --
12.10 0.167 28.55 7.95 36.50 365 9.25 20.60 19.50 10.0 Everything moving
10.70 . 0.148 24.85 5.80 30.65 335 8.45 19.05 18.15 9.5 " "10.10 0.139 22.80 5.00 . 27.80 315 8.0 17.80 17.0 8.8 Heavy bed load
8. 170 0.120 19.40 3.75 23.15 290 7.95 15.65 15.0 7.8 Quickly moving bedjust above crit.
7.70 0.106 15.45 2.70 18.15 255 6.5 12.75 12.15 6.5 Deposit - and.immediate scour
7.60 0.1045 14.65 2.50 17.15 245 6.2 12.15 12.15 6.5 Still squirming,.. pulsating bed
7.90 0.109 13.60 2.35 15.95 235 5.95 11.25 '10.75 5.5 Above cri tical
JCRITICAL I7.30 0.101 10.75 2.60 13.35· 230 5.95 8.15 7.65 4.0 Non-moving bed
6.70 0.092 9.35 1.50 10.85 200 5.1 7.85 7.45 3.8 Flat bed
4.70 0.065 5.40 1.35 6.75 150 3.9 4.05 3.75 2.0 Long flat dunes
2.70 0.037 2.90 1.10 4.00 110 2.8 1.80 1.80 1.0 Long flat dunes
I00\0I
Series G-01-5
1 d'_tel3t section_ .cop rea longs
6h1 . 95. (..L) Aha Aho AhR+6h» Qm v Aha-Aho AhR-l\hD c COMMEN~'SAt m 11\
corrected[in.) [in.] -(in. ] [in.] [gpm] (fps] r· .. [in.] [~~)1.1n. J
2nd Run
3.90 0.191 32.90 10.70 43.60 410 10.45 22.20 20.90 10.5 All suspended
11.40 . 0.158 26.30 6.90 33.20 355 9.0 19.40 18.30· 9.5 " "~0.70 ,0.139 22.25 5.30 27.55 325 8.25 16.95 16.15 8.5 Bed load
8.20 0.113 18.30 4.20 22.50 295 7.5 14.10 13.40 7.0 Slowly moving bed
7.50 0.103 15.40 2.90 '18.30 260 6.6 12.50 11.90 6.0 Pulsating-slidingbed
7.80 0.1075 12.40 2.10 14.50 235 5.95 10;30 9.80 5.0 Bed just slightly.-moving
ICRITICAL I6.70 0.092 10.30 2.60 12.90 220 5.6 7.90 ·7.20 4.0 Non-moving bed
just belowcritical
4.50-
Flat bed, great4.30 0.059 5.80 1.30 7.10 160 4.1 4.20 2.2saltation
3.20 0.044 3.50 1.20 4.70 130. 3.3 2.30 2.00 1.0 Very little ~!!dune buildings
2.80 0.039 2.05 0.35 2.40 95 2.4 1. 70 1.50 0.8 High duneformation
I\CoI
Series G-01-5
i •
/
loop readingstest section-- - ,l:Ih1 . 95
(~) tiliR tJ1n l:IhR+l:IhD Qm V tiliR-l:Ihn b.hR-l:Ihn C COMMENTSl:I-L m m
corrected[in.] [in.] [in.] [in.] [gpm] [fps] (in.] (in.) (%]
3rd Run , .;.:
13.70 0.189 32.00 10.30 42.30 405 10.3 21.70 20.40 10.5 Everything moving
10.80 0.149 24.70 6.30 31.00 340 8.65 18.40 17.40' 9.0 .' II "9.00 0.124 20.05 4.40 24.45 300 7.6 15.65 14.950 7.75 Moving, sliding
bed
7.70 0.106 13:90 2.75:.16.65 250 6.35 11.15 10.55 5.5 Pulsating bed
7.90 0.109 12.10 2.35 14.45 235 5.95 . 9.75 9.25 '5 Just slightlymoving bed
ICRITICAL I6.90 0.095 10.80 2.15 12.95 220 5.6 8.65 8.25 4.25 Just' below
critical,non-moving bed
3.20 0.044 3.45 1.40 4.85 125 3.2 2.05 1. 75 1.0 Flat bed
Series G-Ol-5
{
C = 5.00%CRITICAL CONDITION
Vc = 5.95 fps
di1ti't t'-' es sec ~n oop rea ngs
6hH O' (~~ 6hR
6hD
6hR+6hD ,~ V 6hR-6hD
6hR-6hD C COMMENTS2 ·m m.corrected
[in. ] [in. ] . tin.] [in. ] [gpm] [fps] [in. ] [in. ] [%]
9.40 0~0663 14.50 13.40 27.90 365 9.2 1.10 0.95 0.50 Complete suspension
6.90 0.0486 10.80 10.10 20.90 310 7.9 0.90 0.70 0.35 " "5.00 0.0353 7.65 7.10 14.75 265 6.7 0.55 0.30 0.15 Heavy bed load
3.20 0.0226 4.85 . 4.50,
9.40 220 5.6 0.35 0.20 0.10 Scour fluctuations
3.00 0.0211 {4.55 4.15 8.70} 200 5.1 {0.50 0.30 0.15} Deposits for awhile4.60 ' 4.10 8.70 0.40 0.20 0.10 then slides again
..
1.90 0.0134 2.95 2.85 5.80 165 4.15 0.10 -- -- Infrequent sandslugs; circulatedsystem at high Qand made 2nd run.
3.30 0.0233 5.05 4.25, . 9.30 215 5.45 0.80 0.60 0.30 Heavy bed load
1.65 0.0116 2.65 2.20 4.85 150 3.9 0.45 0.25 O~ 12, ICRITICALI dune for-mation due todistribution
,
I\0NI
Series G-Ol~6
CRITICAL CONDITION {cVc
= 0.12%
= 3.90 fps
'" ,
" . I '
".
n
di1tit t~ esse~~~)n
oop rea ngs
~hH20 ~hR ~hD ~hR+~hD Qm 'I V~hR-~hD ~hR-~hD C COMMENTSM'm ': m
corrected[in. ] [in.] [in.] [in.] , [gpm] [fps] [in. ] [in. ] . [%]
9.90 0.0698 15.35 ' 13.35 28.70 360 9.15 2.00 1.85 0~95 Complete suspensio;
7.10 0.0500 10.85 9.60 20.45 300 7.75 1.25 1.05 0.55 Bed load transport
3.40 0.0240 5.05 4.35 9.40 ' 200 5.1 0.70 0.50 0.25 Pulsating bed
2.85 '0.0201 4.05 3.60 7.65 1'80 4.8 0.45 0.35 0.15 Settling with im-med iate' scour,just above crit.
, ICRITICAL!
2~10 0.0148 3.25 ' 2.80 6.05 160 4.1 0.45 0.30, 0.15 Dune formation---"
I\DWI
Series G-01-7
.,' {
C = 0.15%CRITICAL CONDITION,
Vc ,= 4.65fps
n9.90 0.0698, 15.05 12.45 27.50 355 ' 9 0 00 2.60 2.45 1.25 'Complete suspensio
5.40 0.0381 8.30 ' 7.00 15.30 265 6.70 '1.30 ' 1.10 0.55 Bed load transport
{4.35 3.75 8.10} ,,
{ 0.60 0.40 0.20}2.80 0.0198 4.45 4.00 8.45 200 5.1'0 0.45 0.30 0.15 ICRITICAL(•
. " {c = 0.20%CRITICAL CONDITION "
, , Vc = 5.10 fps
Series G-01-8
di11.. testse(i~)n
oop rea ngs
tihH 0 tihR tiho tihR+ tiho Qm V tihR-tiho tiha-ti~ C COMMENTS. 2 tit m mcorrected
[in. ] [1n. ] . [1n.] [1n. ] [gpm] [fps] [in o ] [in. ] [%]
8.35 0.0589 12.65 9.75 22.40 310 7.9 2.90 2.75 h40 Suspended and bedload transport
5 .. 30 . 0.0374 8.25 6.45 14.70 255 6.45 1.80 1.60 0.80 Pu1sa~ing bedmotion and shear,
4.85 0.0342 6.75 5.30 12.05 225 5.7 1.45. 1.20 0.60 Just above thecrit. condition
4.10 0.0289 5.90 4.65' 10.55 210 5.35 1.25· 1.00 0.50 ICRITICA~1
3.20, 0.0226 5.00 4.25 9.25 190 : 4.8 0.75 0.60 0.30. Sporatic settling,
'.long dunes
3.00 0.0212 3.75 2.9S 6.70 . 160 4.1 0.80 ',0.60 0.30 COijJ.pletelystationary bed
I\0.p-I
Series G-01-9
CRITICAL CONOITION {C
. YC
= 0.50%. ~..
= 5.35 fps
' ..
n
di1i,. testse(1~)n
l( op rea ngs
6hH 0 6hR6h 6~+6hD Qm V 6hR-6h
D 6~-6hD C COMMENTS. 2 M'm D mcorrected
[in. ] [in. ] . [in.] [in. ] [gpm) [fps] [in. ] [in. ] [%J
-13.95 0.0985 21.45 14.90 36.35 400 9.55 6.55 6.30 3.20 Complete suspensio
8.15 0.0575 12.35 9.70 22.05 310 7.9 2.65 2.45 1.25 .. ..7.10 0.0500 10.60 8.30 18.90 290 7.35 2.30 2.10 1.05 .. . ..
9.1C},
6.30 0.0443 7.10 16.20 265 . 6.7 2.00 1.80 0.90 Heavy bed load.condition
5.60 0.0395 8•.00 6.35 14.35 245 6.2 1.65 1.40 0.70 Particles sliding.and becomingvisible
4.80 0.0338 6.95 5.60 12.55 230 5.85 1.35 1.15 0.60 }4.20 0.0296 6.45 5.10 11.50 . 225 5.7 1.35 ·1.10 0.55 ICRITICALI4.50 0~a317 6.70 5.45 12.15 225 5.7 1.25 1.00. 0.50
3.50 0.0247 5.05 4.25 9.30 . 200 5.1 0.80 0.60 0.30 Bottom of depositis non-moving
1.35 0.0095 -1.95 1.50 3.45 llO 3.3 0.45 0.30 0.15· Long dune deposit
I\0IJ1I
Series G-01-10
.. {C '= 0.60%CRITICAL CONDITION
Vc = 5.80 fps
1 di,.. testse~1\)n
oop rea ngs
6hH 0 6hR 6h 6ha+6hD Qm V 6hR-6hD 6ha-6hD C COMMENTS6t m D m. 2corrected
[in. ] [in. ] [in.] [in. ] [gpm] [fps] [in. ] [in. ] [%]~-
16.85 0.1190 ~5 .15 16.10 41.25 440 11.15 9.05 8.80 4~40 Complete suspension
12.05 0.0849 17.25 11.10 28.35 360 9.1 6.15 5.95 3.00 Suspension and bedload. transport
8.60 0.0606 12.05 8.20 20.25 .300 7.65 3.85 3.70 1.85 Heavy bed load ,
transport
6.15 0.0434 8.50 6.20 14.70 255 6.5 2.30 .2.10 l.05 Just above the. crit • condition
JCRITICALI5.45 0.0384 7.65 5.70 13.35 235 5.95 1.95 1.70 0.85 Just into the
-.deposit regime
3.30 0.0233 5.05 ·4.05 9.10 195 4.95 1.00 0.80 0.40 Dune deposit,- . fluctuating head
. loss readingsdue to sporaticduning
I\00\I
Series G-01-11
- {·C '= 1.09%CRITICAL CONDITION ._
Vc - 6.40 fps
". ,
" ." .------- .,/
0.3
IS.S•
15
14.25•
13.0•
0= 4in.d50 =0.88mm
5=-0.060€ =0.00003 ft.
11.2511.25 ••
/
3 4 5 6 7 8 9 10
Vm Ups) t MIXTURE VELOCITY
0.01 L.-_..L.---.1.-.l~-J-_-1-_..L.---1...--.1.---I---I --L... _
2
0.05
0.2
-.en0.090.08
0.07
0.06
0.02
wa:::::::>....x~
_.--=;~ 0.03
~I~, ,
en_.-cno.....J
'. 0«w:I:
\.
Plot of Series G-02 Data
-97-
loop readingste t ection"'"
8 S 1 ----..,. .(.AlL.)
,t.h1. 95 ~h h.hD AhR+~hD Qr.l V ~h ... l.\h ' ~hR"'hhD C Cl~ENTS. M, R In R D. m
corrected[in.) [in. ] [i.n •J [in.J [gpm] [fps] [in.) r J [%]"in.
8.80 0.121 27.35 16.70 44.05 410 10.45 10.65 9.35. 4.75 Everything moving,:.;
7.50 0.103 22.65 13.80 36.45 375 9.55 8.85 7.75 4.0 " "5.20 0.072 13.65 9.40 23.05 300 7.6 4.25 3.55 1. 75 " "4.30 0.059 11.70 7.75 19.45 270 6.85 3;95 3.35 1.6 " "3.50 0.048 8.90 6.25 15.15 235 5.95 2.65 2.15 1.25 Sliding bed
2.90 0.040 7.60 5.50 13.10 225 5.75 2.10 1. 60 0.75 Pulsating bed
2.50 0.034 6.45 4.85 11.30 210 5.35 1.60 1.20 0.6 " "2.00 0.028 5.55 4.10 9.65 190 4.8 1.45 1.05 0.5 Slowly pulsating
3.50 0.048 9.60 6.35 15.95 250 6.35 3.25 2.65 1.3 Everything moving
2.80 0.0385 7.30 5.45 12.75 225 5.75 1.85 1.35 0.75 " "2.40 0.033 6.40 4.75 11.15 205 5.2 1.65 1.20 0.6 Pulsating
2.15 0.0295 5.45 4.20 9.65 190 4.8 1.25 0.85 0.5 ICRITICAL I
I\000I
Series G-02-1
{
c = 0.50%CRITICAL CONDITION
Vc = 4.8 fps
/
loop readingtest il"'ction- c - 8
bh1. 95l:t.h
MlR t.hfJ 6hR+uho\.
(A1i) Ql1l
V AbR-bbn MlR-tlhD c COM·tENTSm~ m
correcterJ[in.) (in.]
,. ,[in.J [gpm) [fps] (in.] [in.J ['oJLill. J
.. '.~._.- .. ,.... ...__. _..._.._'.-
9.20 0.127 25.65 12.60 '. 38.25 400 10.2 13.05' 11.85 6.0 Everything moving
0.110! ..:
8.00 21.85 10.55 32.40 365 9.2 11.30 10.25 5.25 " "7.10 0.098 19.25 9.40 28.65 340 8.65 9.85 8.95 4.5 " "6.10 0.084 15.50 8.0 23.50 300 7.6 7.50 6.70 3.3 " "5.70 0.078 13 .90 7.05 20.95 290 7.35 6.85 6.15 3.0 " "4.10 0.0565 9.85 5.50 15.35 250 6.35 4.35 3.75 2.0 Sliding bed
4.20 0.058 8.80 5.0 13.80 235 5.95 3.80 3.30 1. 75 Just pulsating
3.70' 0.051 7.25 4.55 11. 75 220 5.65 2.70 2.20 1.25 Just above crit.
3.00 0.041 6.05 4.0 10.05 200 5.1 2.05 1.65 0.75 ICRITICAL 12.50 0.0345 3.60 2.0 5.60 145 3:7 1. 60 1.30 0.5 Deposit
2nd Run,
7.00 0.0965. 19.25 9.55 28.80 320 8.1 9.7 8.8' 4.5 Everything moving
5.70 0.078 14.65 7.35 22.0 285 7.25 7.3 6.5 3.3 " "4.90 0.067 11.40 5.85 . 17 .25 255 6.5 5.55 4.95 2.5 " "4.20 0.058 9.05 5.25 14.30 230 5.85 3.80 3.3 1. 75 " "3.50 0.048 7.30 4.30 11.60 210 5.35 3.0 2.6 1.3 : Bed just moving
2.90 0.040 5.85 3.75 9.60 195 4.95 2.1 1.7 1 . JCRITICAL I1.10 0.015 2.25 1.85 4.10 120 3.1 0.4 0.2 0.25 Flat bed
I\0\0I
CRITICAL CONDITION: C = LOO%VC = 5.1fps
Series G-02-2
loop reudings--e t sec tion,..l_ ~s -, .\
~. (_~~1 .) 6hR M1D 6hR+~hD Qm V AhR-bhD 6hR-~bD C COMMENTS{,n1 . 95 .(, nl m-corrected
r- ) (:In.] [in.] [in.) (gpm] [fps] [in.) [in.J (70)Lin.
12.90 0.178 33.00 i1.05 44.-05 410 10.4 21. 95 20.65 10,,5 Everything moving:.;
1t.50 0.158 28.90 8.45 37.35 380 9.65 20.45 19.25 10.0 " "9.10 0.125 21.60 5.20 26.80 315 8.0 16.40 15.60 8.0 " "7.00 0.096 16.65 3.60 20.25 275 7.0 13.05 12.35 6.3 " "6.70 0.092 14.60 2.80 17.40 255 6.5 11.80 11.20 5.75 Sliding bed
6.10 0.084 12.05 2.70 '14.75 235 5.95 9.35 8.85 4.5 Qucik1y pulsating
6.50 - 0.089 10.65 2.45 13.10 225 5.75 8.20 7.70 4.0 Slowly moving,just below crit.
5.80 0.080 9.15 2.30 11.45 215 5.45 6.85 6.35 3.3 ICRITICAL[
4.90 0.0675 7.40 2;0 9.40 190 4.8 5.40 5.0 2.5 Deposit
5.10 0.070 7.70 2.30 10.0 195 4.95 5.40 5.0 2.5 "3.70 0.051 4.90 1.60 6.50 150 3.9 3.30 3.0 1.5 Flat bed
2.40 0.033 - 3.0 1.0 4.0 120 3.1 2.0 1. 70 0.75 " II
1. 70 0.023 1.45 1.0 2.45 90 2.3 0.45 0.30 0.25 Dunes
2nd Run
6.20 0.085 12.40 2.40 14.80 235 5.95 10.0 9.50 4.75 Quickly pulsating
6.30 0.087 9.65 2.40 12.05 215 5.45 7.25 6.85 3.5 Slowly pulsating
.. -
I....ooI
Continued Series G-02-3
~.
Series G-02-3
loop readingstE'1st sectio:l.- .. "-eo:1 l . 95(..M!...) 6h ~hD L\hR-M1D
'\~hR+6hD Qm V L\h
R-6h
D ·C C<WAMENl'S6t R m'm
[in.)corLected
[in.] [in.) (1,n. J [gpnl ] (fps] [in.] (in.) ('7oJ-
5.20 0.0715 8.60 2.05 10.~5 205 5.2 . 6.55 6.15 3.25 ICRITICAL I5.60 0.077 7.95 1.50 9.45 190 4.8 6.45 6.05 3.0 Flat bed
4.80 0.066 6.70 1. 30 .' 8.0 170 4.35 5.40 ·5.10 2.6 II II
4.30 0.059 5.70 1. 70 7.40 160 4.1 4.0 3.60 1. 75 " II
3.80 0.052 4.45 1.60 6.05 145 3.7 2.85 2.55 1.3 II "2;40 0.033 2.95 1.40 " 4.35 120 3.1 1.55 1.25 0.6 . " "L50 0.021 1. 50 0.65 2.15 90 2.3 0.85 0.65 0.3 Dunes
..I.....o.....I
CRITICAL CONDITION {c. Vc = 5.35 fps
.':
test eection_ loop readinga
11.95 (..lill....) t.hRl\hn AhR+~~ . Qm 1/ AhR-~hn AhR-Ahn C COMMENTS
~t m mcorrected
(in.] [in. J (in. J (in.] (811m] [fpa] [in.] (in.) [%)---,.._.- _ ............_ ...-------_.._...._.y...._--_..
10.40 0.143 21.40 -1.05 20.35 270 6.85 22.45 21.85 11.25 Sometimes stopping
9.60 0.132 20.25 -1.65 18.60 260 6.6 21. 90 21.30 11.0 " "8.20 0.113 17.70 -2.0 15.70 240 6.1 19.70 19.20 10.0 " "7.60 0.105 14.90 -2.20 12.70 215 ·5.45 17.10 16.70 . 8.5 Quickly pulsating
8.40 0.116 11.30 -0.95 10.35 200 4.95 12.25 11.85 6.0 ICRITICALI
6.70 0.092 8.30 -1.50 ; 6.80 155 4.0 9.80 9.40 4.75 Flat bed
6.60 0.09i 7.85 -1.60 6.25 145 3.7 9.45 9.15 4.5 " "I
0.0825 135 3.4 " ".... 6.00 . 7.10 -1. 70 5.40 8.80 8.50 4 ..250N
0.052 3.40 105 2.7 3.10 " "I 3.80 3.35 -K>.05 3.30 1.6
2nd Run
16.00 0.221 36.45 5.35 41.80 365 9.2 31.10 30.0 ·15.5 Everything moving
13.80 0.190 31.10 2.50 33.60,
335 8.55 28.60 27.70 14.25 " "11.30 0.156 25.65 -0.10 25.55 290 7.35 25.75 25.05 13.0 " ".-10.00 0.138 21.55 -1.05 20.45 260 6.6 22.60 21. 95 11.25 " "8.00 0.110 18.35 -1.85 16.50 240 6.05 20.20 19.70 10.0 Slowly ~u1sating
7.70 0:106 15.10 -2.30 12.80 215 5.45 17.40 17.0 8.75 " "8.80 0.121 12.0 -1. 70 10.30 195 4.95 13.70 13.30 6.75 ·1 CRITICAL I
fluctuatingwith scour-deposit
4.40 0.061 4.10 -0.60 3.50 100 2.55 4.70 4.40 2.25 Flat bed
Continued Series G-02-4
loop readirgst t "ti,..' es . l'lc.;C on __I •
t,h1 . 95(-PlL) l.ihR bhn llhn.+thn ~I V MlR-~hD llhR-6hn C CO~1ENTS
llt' mm
corrected[in.) (in.) [in.) [in.) (gpm) [fps] [in.] [in.) (%]
-....- ....- ..-3rd run
9.30 0.128 19.95 -2.20 245 22.15) ..-
17.75 6.15 21.65 1l.0 Deposit'scour
7.80 0.107 16.30 -2.90 13.40 205 5.2 19.20 18.80 9.5 " "7.80 0.107 16.15 -2.75 13.40 205 5,2 18.90 18.50 9.5 " "
ICRITICAL(
8.20 0.1l3 11.40 -2.10 .: 9.30 180 4.6 13.50 13.10 7.0 Just deposited,thick bed
7.10 0.098 8.70 -1. 90 6.80 150 3.9 10.60 10.30 5.25 Flat bed
5.40 0.074 6.50 -1.50 5.0 125 3.2 8.0 7.70 4.0 " "4.00 0.055 3.60 -t{).20 3.80 110 2.8 3.40 3.20 1. 75 " "
I....eI
Series G-02-4
CRITICAL CONDITION{
C : 7.,00%
Vc - 5.0 fps
I·
i~.
0.2
Clear Fluid
D =4in.d50=0.45 mm
5=0E =0.00003 ft.
0.01 L.--L..---..L.--...L..-_...L..----I._..L.....__'__~__'__ ______IL.__
2 3 4 5 6 7 8 9 10 15
Vm (fps) , MIXTURE VELOCITY
0.10.09
0.08CJ)CJ) 0.07
9o«w:I:
W0:::J....X~ 0.03
E
~I~i, , 0.02
Plot of. Series G-OOI Data
-104-
loop readi gte t section,.. ~3 - n s
t\\.95(~) t.h
R 6hn ~hR+L\~ Qm V AhR-tlhn tlhR-tl~ C CO}~RNTSA~ m m
corrected(in.) [in. ] [in.] [in..) [gpm] [fps] [in.] r· ] ['7oJl.1.n •
- .7.15 0.098 24.45 14.35 38.80 415 10.55 10.10 8.70 4!~5 Everything moving
5.20 0.0715 17.00 10.10 27.10 345 8.75 . 6.90 6.00 3.10 " "4.10 0.0563 13.15 9.25 22.40 315 8.00 3.90 3.10 1.60 .' " "2.95 0.0405 8.75 5.90 14.65 255 6.45 2.85 2.40 1.20 " "2.65 0.0364 7.90 . 5.45 13.35 240 6.10 2.45 2.00 1.02 Bed particles
'.visible
2.15 0.0296 6.75 4.70 11.45 220 5.60 2.05 1.65 0.85 Pulsating, almostdeposited, justabove critical
1.85 0.0254 5.60 3.95 9.55 200 5.10 1.65 1.25 0.65 ICRITICALI
1. 75 0.0240 5.10 3.70 8.80 190 4.80 1.40 1.00 0.50 Flat bed
1.15 0.0172 3.50 2.70 6.20 160 4.10 0.80 0.45 0.25 " "0.95 0.0130 2.15 1.60 3.75 120 . 3.10 0.55 0.30 0.15 " "
I·.....ol.nI
Series G-001-1
{
c _= 0.65%CRITICAL CONDITION
Vc -.5.10 fps
..'
./
loop readingstest section,." - ;
"ih1 . 95(...£h...) AhR Ahn AhR+Ahn ~1 V AhR-Ahn AhR-Ahn c CONMENTS
A-L m mcorrected
[~ , (in.] [in.) [in.) [gpm] [fps) [in.] (in.) [%j... r•. .i
8.65 0.119 30.45 15.25 45.70 435 10.60 15.20 13.70 7.0 Everything moving;.:
7.35 0.101 25.30 12.35 37.65 395 9.50 12.95 11.65 6.0 " "(400) .'
6.45 0.089 21.20 10.30 31.50 360 9.15 10.90 9.90 5.0 Bed load
5.95 0.082 19.55 9.25 28.80 345 8.75 10.30 9.30 4.75 " "4.95 0.068 16.05 7.55 73.60 315 8.05 8.50 7.70 3.9 Sliding bed
4.05 0.056 12.05 6.20 18.25 275 7.00 5.85 5.35 2.7 " "3.45 : 0.0475 9.60 5.40 15.00 245 6.30 4.20 3.80 2.0 Pulsating bed
(250)
2.75 0.038 7.90 4.70 12.60 225 5.70 3.20 2.80 1.5 Just above critical
{7.85 4.30 12.15} { 3.55 3.15 1.6} rCRITICAL)2.85 0.039 220 5.607.75 4.40 12.15 3.35 3.05 1.5
2.35 0.032 6.20 3.80 10.00 200 5.10 2.40 2.10 1.1 Flat bed
1. 75 0.024 4.55 . 3.00 7.55 170 4.40 1.55 1.25 0.7 " "
It-'o0'\I
Series G-001-2
CRITICAL CONDITION {cVc
= 1~50%
= 5.6 fps
loop readingstest section..,... -bh1 . 95
(~) t\hR t\hD t\hR+t\~ Qm V tiliR-tiliD tiliR-t.hD C COMMENTSM" m m
corrected[in.] [in.] [in.] [in.] [gpm] [fps] [in.] [in.] [%]
-_ ..~ --9.35 0.129 31.90 13.40 45.30 430 10.95 18.50 17.00 8.il5 . Everything moving
8.65 0.119 28.75 11.50 40.25 410 10.45 17.25 15.95 8.2 " "7.45 0.103 23.85 8.80 32.65 365 9.30 15.05 13.95 7.2 Heavy bed load
6.45 0.089 20.00 7.40 27.40 330 8.40 12:60 11.80 6.0 " " "5.55 0.0765 16.40 6.15 , 22.65 305 7.75 10.25 9.55 4.9 " " "5.05 0.0695 14.35 5.35 19.70 285 7.25 9.00 '8.40 4.3 Sliding bed
4.85 0.067 12.25 4.80 17.05 265 6.75 7;45 7.05 3.7 Quickly pulsating
4.55 0.0625 11.45 4.75 16.20 255 6.50 6.70 6.30 3.3 " "(260)
3.85 0.053 9.75 4.25 14.00 245 6.20 5.50 . 5.20 2.7 .I CRITICAL I3.65 0.050 8.90 3.50 12.40 , 225 5.70 5.40 5.10 2.6 Flat bed
3.05 0.042 6.80 3.05 9.85, 200 5.10 3.75 3.50 1.8 Thickening flatbed
2.55 0.035 5.45 2.70 8.15 180 4.55 2.75 2.55 1.3 Sa1ta ting bedload
2.15 0.0295 4.90 2.70 7.60 170 4.40 2.20 2.00 LO Sa1tating bedload
1.45 0.020 2.95 2.00 4.95 135 3.60 0.95 0.80 0.4 Thick bed, littlemoving
_.. '. ---
II-'o'-II
CRITICAL CONDIT ION: C = 3.00%Vc = 6.25 fps
Series G-001-3
10 P e dite t se<:ti on.,.. s 0 r a ngs
t.h l . 95(..ML_> ~hR ~hD ~hR+t.hD Qm V l\hR;'~hD liliR-~hD r. COMMENTS
~.~ mm corrected
[in. J . [in.) [in.} [in.] [gpm] [fps) (in.] (in.] [%]
-9.85 . 0.136 34.25 8.40 42.65 400 10.20 26.15 24.85 1Z ..:7 5 Everything moving
8.55' 0.i175 28.05 5.45 33.50 360 9.15 22.60 21.60 11.1 " "7.15 0.0985 21. 95 3.05 25.0 310 7.90 18.90 18.20 9.3 Mostly bed load
6.15 0.0845 16.00 1. 70 17.7 265 6.75 14.30 13.80 7.0 Slowly moving bed,just abovecritical
ICRITICALl
5.75 .0.079 13.80 1. 70 15.5 245 6.20 12.10 11.70 6.0 Just belowcritical,thickening bed
6.45 0.089 16.05 1.80 17.85 270 6.85 14.25 13.75 7.0 Just belowcritical,thickening bed
5.75 0.079 13.50 1. 05 14.55 235 5.95 12.45 12.05 6.2 Deep flat bed
5.55 0.0765 12.20 0.90 13.10 225 5.70 11.30 10.90 5.6 " " "5.05 0.0695 10.05 '0.70 10.75 200 5.10 10.35 10.05 5.2 " " "4.25 0.0585 7.60 0.85 8.45 175 4.55 6.75 6.45
.3.3 Still suspension
load
3.45 0.0475 5.65 1.00 6.65 155 4.05 4.65 4.35 2.25 Saltation load
2.85 0.039 4.30 0.80 5.10 130 3.45 3.50 3.30 1.7 Flat bed
1.65 0.023 2.35 0.85 3.20 110 3.00 1.50 1.30 0.7 " "
•~o00
•
CRITICAL CONDITION: C = 7.00%VC = 6.5 fps
Series G-001-4
n
di1tit t,.. es sec ~n' oop rea ngs
AhH O' (~ l1hR AhO . AhR+AbOQ . Vm l1hR-Ahn AhR-l1hn 'C COMMENTS2 m m
.tin.]corrected
[in.] [in. ] [in. ] [gpm] . [fps] [in. ] [in. ] [%]...-
9.65 0~0681 14.90 13 .80 28.70 375 9.55 L10 1.00 0.50 Complete suspens:Lo'
7.10 0.0501 10.90 10.30 21.20 310 7.9 0.60 0.45 0.22 " "
5.00 ·0.0353 7.80 7~30 15.10 265 6.7 0.50 0.30 . 0.15 Susperision and bedload transport'
4.20 ·0.0296 . 6.55 6~20 12.75 245 6.2 0.35 0.15 0.07 Heavy bed load
3.15 0.0222 5.00 4.70' 9.70 . 215 5.45 0.30 . 0.10 0.05 Sliding bed load
f"45 4.10 8.55} f"35 0.15 O.Ol} Just above crit.2.85 0.0201 4.55 4.05 8.60 195 4.95 0.55 0.35 0.15 with s'poratic
4.35 4.10 . 8.45 0.25 0.10 0.05 bed scour at.. partially closed
valve
"2.20 2.00 4.20} .r- 2O 0.10 0.05} Persisten~ scour-1.35 0.0095 2.15 1.85 4.00 135 3.6 Q.30 0.15 0.07 ing, sometimes
,2.15 1.95 4.10 0.20 0.10 0.05 critical
ICRITICAL I0.65 0.004~ 1.10 0.90 2.00 100 . 2.55 0.20 0.10 0.03 Stationary deposit
I....a~I
Series G-001-5
= 0.05%
". \
,{CCRITICAL CONDITION
. V .~ 2.75 fpsC
di1i~ test ser Un oop .rea nga6 . 6h 6h
R6ho 6h
R+6ho Qm VhH 0 . 6t 6h
R-6ho 6hR-6hO' C COMMENTS2 m m
.corrected[in. ] [in. ] [in.J [in. J [gpm] [fps] [in.J [in. J . [%J
8.95 . 0.0632 14.00 12.50 26.50 355 9.0 1.50 1.30 0.65 Complete suspension
7.05 0.0497 11.20 10.00 21.20 310 7.9 1.20 '1.05 0.55 " "5.15 :0.• 0363 8.20 7.40 15.60 . 265 6.7 0.80 0.60 0.30 Suspension and bed
load transport
4.10 0.0289 6.50 5.90 . 12.40 230 5.85 0.60 0.40 0.20 Heavy.bed load
3.20 0.0226 5.00 4.65 9.65 215 5.45 0.35 0.20 0.10 Slid ing bed ..3.10 :: 0.0219 {4.90 4.30 9.20} 200 5.1 { 0.60 0.40 0.2~J Sporadic scouring
4.80 4.35 9.15 .' 0.45 0.25 ' 0.12 and deposit..
1.80 0.0127 2.90 2.55 5.45 160 4.1 0•.35 0.20 O.Itl ICRITICAL]
1. 75 0.0124 2.80 2.45 5.25 150 3.9 0.35 : 0.15 0.08 Just below crit.
It-'t-'oI
.,'Series G-001-6
..' \
, . j ,.' {c = 0,.10%CR1TICAL CONOITION "
, V' = 4. 10 fps,. • C
, '
" .. ' ,
1t1t t,. esset6~)n
oop rea ngs6hH 0 6hR 6h
D 6hR+6hD Qm V 6hR-6hD 6~-6hD C COMMENTS2 6t m m
corrected[in. ] [in. ] . [in.] [in. ] [gpm] [fps] [in. ] [in. ] [%]
9.30 0.0656 14.65 12.60 27.25 365 9.2 2.05 1.95 1.00 Complete suspen~ion
6.95 0.0490 10.80 9.50 20.30 310 7.9 1.30 1.20 0.60 " "5.25 0.0370 8.25 7.30 15.55 265 6.7 0.95 0.80 0.40 Heavy 'bed load with
saltation into'less denselypopulated areas
.. of the cross-section
4.10 0.0289 6.30 5.70 12.00 230 5~85 0.60 0.40 0.20 Thickening bed ofsliding particles
3.20 0.0226 e· 95'
4.15 9.10} 200 5.1 { 0.80 0.55 0.25} Pulsating condi-4.90 4.30 .9.20 0.60 0.40 0.20 tions, just above
critical
ICRITICALI1.80 0.0127 2.70 2.45 5.15 155 3.95 0.35 0.20 0.10 Sufficiently below
crit. settling
..............I
Series G-001-7
{
c = 0.20%CRITICAL.~ONDITION _
. VC.- 4.80 fps
n
di1i,. testset~~)n
oop rea ngs6hH 0 6hR ' 6hD
6hR+6hD Qm V 6hR-LihD
6hR-LihD C COMMENTS, 2 Lit m mcorrected
[in. ] [ill. ] [in. ] , [in. J [gpm] [fps] [in. ] [in. ] [%]
9.40 0~0663 14.65 10.85 25.50 355 9.0 3.80 3.65 i.'85 Complete suspensio
7.10 0.0500 11.00 8.50 19.50 310 7.9 2.50 2.35 1.20 Suspension withnot iceab 1e bedload .,
5.35 0.0378 8.25 6.75, 15.00 265 6.7, 1.50 . 1.30 0.65 Heavy bed load
4.35 0.0307 6.60 5.60 12.20 240 6.1 l.00 0.80 0.40 Sliding bed, in-.creasing depositdepth
3.60 0.0254 5.50 4.75 , , 10.25 220 ' 5.6 ,0,75 0.60 0.30 Just above crit.~ondition
3.20 0.0226 4.90 4.20 9.10, , 215 '5.45 0.70 0.50 0.25 ICRITICALl3.10 0.0219 4.60 3.9,0 8,.50 ' 200 5.1 0.70 0.45 0.20 Deposit building
2.20 0.0155 3.40 ~.OO 6.40 180 ' 4.6 0".40 0.25 0.15 " "
I..........,N
I
Series G-001-8
. {c ,= 0.30%CRITICAL CONDITION _
, Vc - 5.45 fps
t., ,
Ser~es G-001-9
di1tit.. tesse(6~)n
oop rea ngs6hH 0 6hR
6hn 6hR+6hn Qm VM~m m 6h
R-6hn 6hR-6hn C COMMENTS. 2
corrected[in. ] [in. ] . [in.] [in. ] [gpm] [fps] [in. ] [in. ] [%]
11.20 0.0790 17.85 10.90 28.75 355 9.0 6.95 6.80 3~40 Total transport
9.00 0.0635 14.25· 8.50 22.75 310 7.9 5.75 5.60 2.80 II II
6.25 0.0441 9.55 6.30 15.85 265 6.7 3.25 3.00 1.50 Heavy bed load .5.30 0.0374 7.95 5.70 13.65 235 5.95 2.25 2.05 1.05 Pulsating bed
4.85 0.0347 7.05 5.05 12.10 225 5.7 2.00 1.85 0.95 )CRITICALI
3.55 0.0250 5.20 3.95 9.15 205 5~2 1.25.
1.10 0.55 Stationary bedIt-'t-'UJ.1
. .: . { C = 1. 00%. CRITICAL. CONDITION _
, .Vc,- 5.70 fps
)' "
'., ,
d1t1t t,. esse(6~)n
oop rea 1ngs
6hH 0 6t m6h
R6ho 6h
R+6ho Qm V 6h
R-6ho 6hR-6h
O C COMMENTS. 2 m
[in. ] . [in.]corrected
[ in. ] [in. ] [gpm] [fps] [in. ] [in. ] [%]
12.10 0~0853 19.20 10.70 29.90 375 9.55 8.50 8.40 4.20 Full suspension
8.80 0.0621 13.70 7.65 21.35 310 7.9 6.05 5.90 2.95 " II .6.50 0.0459 9.95 5.95 15'.90 265 6.7 4.00 3.80 1.90 Heavy 'bed load
,6.25 0.0441 9.00 5.60 14.60 260 6.6 3.60 3.40 1.70 Sliding deposit
5.75. 0.0405 8.45 5.45 13.90 250 6.35' 3.00 2.80 1.40 'Approaching crit •.
5.35 0.0377 7.40 4.95 12.35 230 5.85 2.45 . 2.30 1.15 ICRITICALl'
3.55 . 0.0250 5.20 3.70 8.90 210 '5.35 1.50 1.30 0.65 Stationary deposit
Series G-00I-10
. .' '. {c = 1.20%CRITICAL. CONDITION' ._
. VC'- 5.85 fps
...
,.'.,- ,-- :~..-'
0.2
15
o=4in.dso=0.45mm
S =-0.060€ =0.00003 ft.
3 4 5 6 7 8 9 10
Vm (fps) , MIXTURE VELOCITY
0.01 L...-_.L.---L__....L..._-L-_..I.--.....L.--L---L---L --'--
2
0.05
0.06
0.03
0.10.090.08
0.07
0.04
0.02
eneno-Jo<tLaJ:x:LaJ0::::>I-.~
~
Ei •
.cl~<]<], I
plot of Series G-002 Data
-115-
1 diest section_ oop rea ngs ,6h1 . 95
( ~h ) llhR6hD 6hR+b.hD ~ V 6hR-6hD 6hl'-AhD C COMMENTS
M~m m .\.
corrC!cted[i.n,) [in. J [in.] [in.] [gpll1 ] [fps] [in.] [in. J (x.)
- _..- - ...
2.75 0.0378 8.75 7.50 16.25 265 6.7 . 1.25 0.75 0.40 Everything moving".J.'-
1. 95 0.0268 6.30 5.35 11. 65 230 5.85 0.95 0.55 0.30 Ii "" if
1. 65 0.0227 5.05 4.45 9.50 200 5.1 0.60 0.20 0.10 " ",1.25 0.0172 3.90 3.45 7.35 170 4.35 0.45 0.10 0.05 Rapid Pulses
0.95 0.0130 2.80 2.40 5.20 140 3.6 0.40 0.10 0.05 Deposit whenenoughisand
ICRITICAL [:
1. 07 . 0.0147 3.20 2;70 5.90 155 3.95 0.50 0.15 0.08 Deposit
1.00 0.0137 3.20 2.75 5.95 160 4.1 0.45 0.10 0.05 Deposit mostly inlarger pipe
1:'
t..........at
Series G-002-1
CRITICAL CONDITION= 0.05%
= 3.7 fps
- di1tit t.,.-. 'e s sec on oop 'rea ngs
llb1 . 95(..Ah..) bhR bhn AhR+.\hD
Q V bhR-Ahn ~hR-6hn C COMMENTS, ·At m m
m corrected1-:----
[in.. J [in.] [in.] [i.n.] [gpm] [fps] [in.J [in. J [~O
-- ...~.-... _._- -
3.00 0.0405 9.10 7.55 16.65 270 6.85 .1.55 1.05 0.;~5 Everything movingi·;
2.10 0.0288 6.25 5.30 11.55 220 5.65 0.95 0.50 0.25 " "1. 70 0.0233 5.00 4.45 9.45 200 5.1 0.55 0.10 0.05 " "1.40 0.0192 4.25 3.60 7.85 180 4.8 0.65 0.30 0.15 Very slowly moving
1.10 0.0151 3.00 2.50 5.50 150 3.9 0.50 0.20 0.10 Deposit ICRITICALI
0.90 0.0124 2.35 2.10 ~' 4.45 135 3.45 0.25 - - Flat bed, no movingconcentration
0.20 : 0.00274 0.65 0.65 1.30 65 1.65 0.0 - - Small dunes
2nd Run
3.15 .0439 10.20 8.05 18.25 280 7.35 3.15 2.60 1.33 Everything moving
2.65 .0369 8.25 6.80 15.05 250 6.35 1.45 1.00 0.52 " "2.25 .0313 7.10 5.90 13.00 230 5.85 1.20 0.80 0.42 " "1.95 -..0271 6.05 5.10 11.15 215 5.45 0.95 0.55 0.30 " "1. 70 .0237 5.30 '4.50 9.80 200 5.1 0.80 0.40 0.20 Pulses
1.55 .0215 5.00 4.20 9.20 195 4.59 0.80 0.40 0.20 Particles visible
1.48 .0206 4.50 3.75 8.25 185 4.75 0.75 0.40 0.20 " "0.95 .0132 3.10 2.60 5.70 150 3.9 0.50 .0.20 0.10 Almost, deposit
!CRITICALI
1.05 . .0146 3.10 2.45 5.55 150 3.9 0.65 0.30 0.15 Deposit
II-'t-'........I
Series G-002-2
Continued
1 Ii, est s~ction-r - ,oop reae ngs"'\
6h1 . 95Ah Aha nhn AhR+~hD 0. Vm Mla-Aho Aha-Ahu C CONHENTS( At. ~ m, m
corrected[in.) [in.) (in.] (in.J [gpm] (fpsj [in.] [in.) ['7oJ
.-
1. 05 .0146 3.15 2.50 5.65 150 3.9' 0.65 0.30 0{.}.5 ~Deposit, bed le~s
1. 05 .0146 3.15 2.60 5.75 ' 150 3.9 0.55 0.20 0.10 thick
0.70 .0097 2.00 1.65 3.65 115 2,.95 0.35 0.05 0.02 Deposits awhilethen washes away
0.35 .00049 1.20 0.95 2.15 90 2.3 0.25 0.05 0.02 Single dunes
"
Series G-002~2
I........00I
CRITICAL CONDITION= 0.10%
= 3.9 fps
loop readingste t sec'tion.~ 6
(..Ah...),
bhl. 95 AhR ~hD l\hR+A~ ~n Vm llhR-lu'ln l\hRwAhD C COMMEN'fSM~ m
cOl'rected[in..J [in.J [in.) (in.] [gpmJ [fpa] (in.] [in.] [%]
f--
4.35 0.0597 14.00 8.35 22.35 315 8.0 ' 5.65 4.95 2.55 Everything movingi:
3.45 0.0474 11.05 6.60 17.65 275 7.0 4.45 3.95 2.00 " "2.55 0.0350 7.55 5.20 12.75 225 5.75 2.35 1. 95 1.00 Slowing down,
bed particlesvisible
1. 95 0.0268 5.50 4.05 9.55 200 5.1 1.45 1.10 0.55 Pulsating bed
2.15 0.0295 3.85\
1955.50 9.35 4.95 1.65 1.30 0.65 Pulsating slowly
2.05 0.028~} 5.30 3.70 9.00 190 4.8 1.60 1.30 0.65Deposits,
1. 95 0.0268 then slides
1. 55 0.0212 4.20 3.35 7.55 . 175 4.75 0.85 0.50 0.25 Deposit
JCRITICAL [
1.35 0.0185 3.45 2.65 6.10} {O.SO 0.50 0.25 Deposit,1.55 0.0212} 155 3.95 bed thickens1.45 0.0199 3.45 2.60 6.05 0.85 0.55 0.30
1.15 o.0158J 2.75 2.15 4.90 140 3.6 0.60 0.45 0.25Deposit,
1.05 0.0143 first thinner,then thicker
0.95 0.0130 1. 90 1.25 3.15 110 2.8 0.65 0".45 0.25 First flat bed,then dunes
0.40 ' 0.0055} 1.20 0.95 2.15 90 2.3 0.25 0.05 0.03I' long dunes
0.60 0.0082
,........\0,
CRITICAL CONDITION: C = 0.25%Vc = 4.5 fps
Series G-002-3
e ding1t:lt t.,..i es sec .on_ oop r a s
"'~hl.95 (~) ~hR AhD hhR+l\hD Qm V A.~R-~hD liliR-l1h
DC COMMENTSt.t m
m corrected(in.) [i.n.1 [. 1 [in. ] [gpm] [ips] [in.] [. , [%Jloll ... 1.rI.• oJ
3.00 0.0411 9.00 6.10 15.10 255 6.5· 2.90 2.45 1.30 Everything moving; .;
2.60 0.0357 7.75 5.50 ·13.25 240 6.1 2.25 1.85 0.95 " "2.40 0.0327 6.95 5.05 . 12.00 225 5.75 1. 90 1.50 0.80 Rapid pulses
2.10 0.0288 5.90 4.40 10.30 210 5.35 1.50 1.10 0.57 Slow pulses,I.
bed particlesvisible
..1.90 0.0261 5.50 4.20 9.70 205 5.2 1.30 0.90 0.47 Very slow pulses,
almost deposit
2.10 0.0288 5.30 3.90 9.20 198 5.1 . 1.40 1.05 0.552.10 0.0288 5.50 3.75 9.25 200 5.1 1. 75 1.30 0.67 '> Deposit ICRITICALl1. 90 0.0261 5.30 4.05 9.35 200 5.1 1.25 0.85 0.45
1.87 0.0357 5.00 3.70 8.70 190 4.8 1.30 1.00 0.50 Deposit
1. 90 0.0261 4.90 3.65 8.55 185 4.75 1.25 0.90 0.47 Deposit, pulsating
1. 65 0.0226 4.45 3.40 7.85 175 4.7'5 1.05 0.75 0.40 Deposit
1.05 0.0144 2.60 2.10 4~ 70 140 3.6 0.50 0.20 0.10 "0.55 0.0075 1.40 1.30 2.70 95 2.45 0.10 - - II. long dunes
forming
II-'NoI
Series G-002-4
CRITICAL CONDITION {c.. .. Vc
= 0.55%
= 5.1 fps
. I
loop readingstest sec tion-' -- '"6h COH~ENTSL\h1. 95 EAt.> Aha Ahf) AhR+l\ho Qm Vm tlh -Ah AhR-Ahu CR 0m corrected[in.) [il\. ] (in.J [in.J [gpm] [fps] [in.] ... J [%]L:tn •
... . ~-_.- ."-'" ~. -..._....-- .... . -
5.15 0.0707 15.50 4.90 20.40 290 7.35 10.60 10.00 5.15 Everything movingf·.:
4.80 0.0658 13.20 4.45 17 .65 275 7.0 8.75 8.20 4.20 II II.
4.70 0.0644 11.15 4.05 15.20 250 6.35 7.10 . 6.60 3.40 Particles visible
4.30 0.0590 10.45 3.85 14.30 240 6.1 6.60 6.25 3.20 Slow bed motion
3.70 0.0508 8.65 3.60 12.25 225 5.75 5.05 4.65 2.40 Slow pulsating,almost deposit
!CRITICAL I3.55 0.0487 8.00 3.35 11.35 215 5.45 . 4.65 4.30 2.20 Deposit
3.15 0.0431 6.70 3.10 ~~9 .80 200 5.1 3.60 3.20 1.65 Flat bed
2.65 0.0364 5.70 2.90 8.60 190 4.8 2.80 2.45 11.25. II II
2.75 0.0378J 5.25 2.50 7.75 165 4.2 2.75 2.40 1.22 II II
2.55 0.0350.
2.20 0.0302 4.10 2.35 6.45 160 4.1 1. 75 1.50 0.75 II II
1. 65 0.0226 2.70 1.90 4.60 130 3.3 0.80 0.50 0.25 II II
0.65 0.00895 0.70 0.60 1.30 65 1. 65 0.10 - - l' long dunes,no movingconcentrations
Series G-002-5
{
c =_ 2.25%CRITICAL CONDITION
Vc - 5.5 fps
1 p e <1itit t"r- "es' sec. on_ 00 r a ngs
~hl.95(..a!!...) AhR AbD . hhR+AhD ~l V UhR-AhD AhR-bhn C . COMMENTS
A'~ mm corrected..
[in.) [11.1.] [in.] [in. ) [gpm] [fps] [in.] [in.] [%)
_. - - - f4.70 0.0645 13.60 4.45 18.05 270 6.85 9.15 8.60 4 ..40 Everything moving
.: .;
4.50 .0626 11.00 3.90 14.90 250 6.35 7.10 6.65 3.45 Bed particles oftenvisible, strongpulses
4.00 .0556 9.65 3.70 13.35 230 5.85 5.95 5.50 2.85 Particles visibleslower pulses
3.80 .O530}]CRITICAL!3.70 .0515 8.80 3.50 12.30 225 5.7 5.30 4.90 2.50 Deposit
3.90 .0542
. 3.60 .0500 8.40 3.40 11.80 220 5.25 5.00 4.60 2.35 Flat bed
3.00 .0417}c ~
6.65 3.00 9.65 195 4.95 3.65 3.30 1.68 " "3.10 .0432
2.57 .0358 5.35 2.70 8.05 .,175 4.4 2.65 2.30 1.18 " "3.00 .0417 5.50 2.30 7.80 '170 4.35 3.20 2.85 1.45 I
2.60 .0362 5~35 2.50 7.85 170 4.35 2.85 2.50 1.25 > II II
2.80 .0390 5.45 .2.40 7.95 175 4.4 3.05 . 2.80 1.42
2.30 .0320 5.00 2.05 7.05 165 4.2 2.95 2.65 1.35 II "2.40 .0334 4.90 2.00 6.90 160 4.1 2.90 2.60 1.32
1.60 .0222 2.90 1. 70 4.60 130 3.30 1.20 1.00 0.50 " II ..
I.....NNI
CRITICAL CONDITION: C = 2.50%Vc = 5.1 fps
Series G-002-6
\,\...
'.'-_._- "---
. '1
0.2
153 4 5 6 7 8 9 10
Vm Ups) I MIXTURE VELOCITY
,
5.0 6,5,.... ........ " 3! 14.~0~'0
" /3,1 03.5. ...... 3.0 29
2.t 0:~J,2~2~~'604'2,/ /J~25 2~t
~9/O~'I 1'=:19o:i~ 0.8 0/001.25 ,.°0,1.5 ..-.0':- 'r-4 -~Ol.l 01.1
• 0 1.3
~O·~_/:o.:o-:soo.a -+
0.6 0.9' Ll 'o .Ita
0.' ioj60.6
~: j.0.4 0= 6 in.
dso=0.88mm5=0E= 0.00016 ft.
I0.01 L-_L-......L.__..L-_..L----I._..L.---L-...:..L..--L- L..
2
0.04
0.02
0.05
0.03
OJ0.09
0.08
0.07
0.06
fJ)fJ)
9o«w:::cwa:::::>....X-::E
E
~I~II I
Plot of Series BS-01 Data
-123-
/
loop readingstit t_ es, sec on ', --v( ,'")(~)
/ \
~1.95 ' AhR Aho ' A~+A~ ~', Vm ,-_~~AhD (AhR-AhD I C COMMENTSA~ m
corrected I
[in.] [in.] [in.] [in.] [gpm] [fps] , [in.] [in.] [%] I
0.0480"
3.54 64.2' 55.0 119.2 750 8.50 9.2 3.1 l.q; Everything moving3.05 0.0420 53.8 46.8 100.6 680 7.75 7.b 2.2 1,1 " "2.82 0.0388 46.6 41.0 87.6 640 7.30 5.6 1,6 0.8 Heavy bed load2.79 0.0385 40,6 35.5 '76.1 595 6,75 5.1 1,6 0.8 Sliding bed
jCRITICAL I2.54 0.0349 35.3 30.9 c 66.2 550 , 6.25 4.4 1.'6 0,8 Just below
cri tical2.36 : 0.0324 32.8 28.5 " 61,3 530 6.05 ' 4.3 1,6 0.8 Deposit2.05 0.0282 26.2 ' 23,2 49.4 I 470 , 5.35 3.0 1,2 0.6 Flat bed
0.0183 36.2,1,33 19.2 17,0 410 4.65 2.2 0.8 0.4 " "-
It-'N.p-I
Series BS-Ol-1
.{C' = 0.80%CRITICAL CONDITION V __
6.40 fpsC ,
..
./
di1t't t,. es sec loon oop rea ngs
&"11 . 95(~) , Ah
RAh . AhR+A~ ~. V AhR-A~ . AhR-AhD C COMMENTS
A~ m D mcorrected·
[in.] [in. J [in.] [in.] [gpm] [fps] [in.] [in.} [%]
.3.90 0.0535 68.2· 57.0 125.2 750 8.50 11.2 5.1 2.p, Everything moving
3.74 0.0515 62.4 52.6 115.0 725 8.25 9.8 4.2 2.2 " "3.54 0.0487 55.2 46.7 101.9 680 7.75 8.5 3.7 1.9 Heavy bed load
3.54 0.0487 48.2 41.0 89.2 630 7.20 7;2 3.3 1.7 Sliding bed
3.46 0.0477 44.0 37.4 81.4 615 7.00 6.6 3.0 1.5 Just abover critical
3.34 0.0459 42.4 37.0 79.4 600 6.80 5.4 2.0 1.0 Just abovecritical
3.26 0.0448 41.4 35.8 77 .2 590 6.70 5.6 2.2 1.1 [CRITICAL1\
2.87 0.0394 36.6 31.0 67.6 555 6.30 5.6 2.6 1.3 Thin bed
2.28 0.0314 28.4 24.6 53.0 480 5.45 3.8 1.8 ·0.9 Flat bed
1.67 0.0229 ·21.6 19.0 40.6 425 4.85 2.6 1.2 0.6 " "" "1.59 0.0219 17.0 15.2 32.2 37~ 4.25 2.2 1.1 0.5
II-'
.~I
Series BS-01-2
. {C ::: 1.10%CRITICAL CONDITION .
. VC ::: 6. 70 fp s
/
10 p e dirttiot t,. es sec n_ o r a gs
(-Ah...;,
tlh1. 95 ' t1hR tlhD t1hR+t1~ ~. V liliR-t1hD liliR-t1~ C COMMENTStl,f, m m
corrected[in.) [in. ] [in.. J [in.] [gpm] [fpsJ [in.] [in.J (70)
.__ . ...
4.75 0.0653 65.8 . 52.2 118.0 725 8.25 13.6 8.1 4.2 Everything moving4.38 0.0604 56.8 45.3 102.1 670 7.60 11.5 7.0 3.6 Pulsating bed,
just abovecritical
4.87 0.0670 52.4 42.6 95.0 650 7.35 9.8 5.6 2.9 Just above. critical
4.76 0.0656 50.6 40.8 : 91.4 635 7.25 9.8 5.6 2.9 ICRITICAL]4.28 0.0589 44.0 36.4 . 80.4 600 6.80 7.6 4.2 2.2 Thin bed3.92 . 0.0540 38.6 31.4 70.0 550 6.25 7.2 4.3 2.25 Flat bed3.36 0.0462 29.4 24.9 54.3 .485 5.50 4.5 2.5 '. 1.25 " "
" "2.66 ·0.0368 24.6 21.2 45.8 445 5.05 3.4 1,8 0.9
.
It-'N0\I
Series BS-01-3
{
c = 3.00%CRITICAL CONDITION
Vc = 7.25 fps
din1i(test sect on oop rea gsLih
LihR AhD AhR+AhD ~ V LihR-AhD LihR-LihD C COMMENTSh1 95 (~) m• m corrected
in. ] [in. ] [in. ] [in. ] [gpm] [fps] [in. ] [in. ] [%]-,5.54 0.0762 59.0 45.2 104.2 700 7.95 14.8 9.8 5.q., Pulsating
5.56 0.0765 57.0 44.1 101.1 690 7.85 12.9 8.2 4.2 Just aboutcritical
5.56 0.0765 54.1 41.4 95.5 670 7.60 12.7 8.5 4.35 ICRITICALI
5.33} 0.0738 50.7 39.7 90.4 655 7.45 11.0 6.8 3.5 Deposit5.39
5.23 0.0720 45.7 35.9 81.6 615 7.00 9.8 6.1 3.1 Deposit
1!..66 . 0.0642' 39.7 31.0 70.7 565 6.45 8.7 5.7 2.95 "1!..15 0.0571 33.6 26.6 60.2 525 6.00 7.0 4.5 2.25 Flat bed
b.51 0.0484 28.4 22.6 51.0 480 5.55 5.8 3.8 1.9 " "~.98 0.0410 24.1 19.8 43.9 445 5.05 4.3 2.7 1.4 " "12.26 0.0310 17 .6 15.2 .32.8 385 4.40 2.4 1.2 0.60 Flat bed,
sa1tating
1.64 0.0260 13.4 11.5 24.9 325 3.70 1.9 1.1 0.5 Flat bed,sa1tating
tI-'N.....t
Continued Series BS-01-4
di1ti[ tes sec on oop rea ngs
~hL 95, (~) l\hR Ahn l\hR+Ahn ~ Vm AhR-Ahn AhR-Ahn C COMMENTS, l\t
m correctedin. ] [in. ] [in.} [in. ] [gpm] [fps] [in. ] , [in.] [%]
2nd Run6.05 0.0834 69.5 51.4 120.9 720 8.20 18.1 12.6 6.5 Heavy bed load
5.68 0.0783 64.2 47.4 111.6 685 7.80 16.8 12.0 6.1 Sliding bed
5.59 0.0769 62.0 46.3 108.3 680 7.75 15.7 11.0 5.7 II II
'6.07 0.0835 55.4 ,41.6 - 97.0 650 7.40 13.8 9.7 5.0 jCRITICALI
5.64 0.0775 47.4 36.8 '84.2 600 6.85 10.6 7.2 3.7 Deposit bed
4.87 0.0670 35.8 27.5 ' 63.3 530 6.05 8.3 5.8 3.0 Thick flat bed
LOS : 0.0145 5.8 4.8 10.6 210 2.40 1.0 0.6 0.3 DunesI....N00 'I
Series BS-01-4
, {c = 5.00% 'CRITICAL CONn ITION ' ,
Vc = 7.40 fps
- ;·r
0.2
15
4.8%
. 0=6 in.d50 =0.88 mm
, 5=0.027E=0.00016 ft.
p.&"j .0.25
3 4 5 6 7 8 9 10
Vm (fps) I MIXTURE VELOCITY
0.01 '--_..L-___L~-...1.---L--.L.---L..---L---L---L-----L..
2
0.05
0.10.090.08
0.07
0.06
0.04
Cf)Cf)
o....Jo<tLLI::L
0.03E
", I.r:.1'(. <l <l, ,
Plot of Series BS-03 Data
-129-
Series BS-03-1Continued
10 p eadingte t ection,.. B S - 0 r .S
(~),
t.h1 . 95 · Aha Aho AhR+AhO Qm Vm AhR-Ahn AhR-Ahn C COMMENTStit m
-corrected[in.) [in.] '(in. ] [in.) [gpm] [fps] [in. J [in.] [%]
3.22 0.0440 54.0 46.4 100.4 675 7.70 7.6 2.9 1.5 Everything moving
3.05 0.0420 49.4 42.5 91.9 645 7.35 6.9 2.8 1.4 " "2.92· 0.0401 43.7 37.9 81.6 610 6.95 5.8 2.3 1.2 Heavy bed load
3.00 0.0412 40.5 35.2 75.7 585 6.65 5.3 2.1 1.1 Pulsating, justabove critical
2.82 0.0388 36.4 32.0 - 68.4 560 6.35 4.4 1.5 q.75 rCRITICALlI
2.74 0.0377 33.4 29.2 62.6 540 6.15 4.2 1.5 0.75 Deposit,thin bed
2.13 0.0292 27.0 23.9 50.9 485 5.50 3.1 1.1 0.6 Flat bed
2.05 0.0282 21.4 19.0. 40.4 440 5.00 2.4 0.9 0.5 " "1.31 0.01800 16.8 15.1 31. 9 380 4.35 1.7 0.5 0.25 " "
2nd Run
3.95 0.0542 61.4 52.4 113.8 . 730 8.30 9.0 3.2 1.6 Everything moving
3.08 0.0422 50.8 43.6 94.4 660 7.50 7.2 2.8 1.4 " "2.97 0.0394 45.2 39.1 84.3 625 7.10 6.1 2.3 1.2 Heavy bed load
1.95 0.0405 41.8 36.1 77.9 595 6.80 5.7 2.3 1.2 Pu1sa ting, justabove critical
2.82 0.0387 39.0 33.8 72.8 570 6.50 5.2 2.2 1.1 ICRITICAL)
2.54 0.0349 33.6 29.1 62.7 530 6.05 4.5 2.0 1.0 Deposit
2.49 0.0342 30.4 26.4 56.8 500 5.70 4.0 1.8 0.9 Thin bedI
II-'WoI
;/
/
1001" readingste t section.. s-~
.~h1. 95 (--eh...) AhR Ahn A~+Ahn ~ V AhR-Al~ AhR-Ahn C COMMENTSb-L m m
-- correc ted. in.) [in.] [in.] [in.] [gpm] . [fps] [in.) r· ] [%].. lon.
2.10 0.0289 28.2 24.6 52.8 485 5.50- 3.6 1.6 0,.8 Flat bed,: .,'
1.69 0~0232 ·24.0 21.2 45.2 455 5.15 . 2.8 1.2 0.6 . " "1.59 0.0218 20.4 17.6 38.0 410 4.65 2.8 1.3 0.6 " "1. 36 0.0187 16.85 14.7 31.55 370 4.20 2.15 1.0 0.5 " "
Series BS-03~1'
. {cCRITICAL CONDITION
. . Vc
= 1.00%
= 6.40 fps
. ~ .J ','
, I.:., I}.
,,1,
:-
di1i.. test sect on--., oop rea ngs ,b.hl. 95· (...AL) , l\h
Rl\h . l\hR+l\~ ~. V l\hR-b.hD
. l\hR
-l\hD C COMMENTS
M, m D ·mcorrected
(in,) [in. ] -(in. ] [in.] [gpm] (fps] (in.] [in. ] [%]
- ..- ... - ..-, , '.
4.77 0.0655 59.5' 47.9 107.4 720 8.20 . 11. 6 6.1 3.2 Everything moving
4.83 0.0662 56.15 46,.0 102.15 705 8.00 10.15 4.95 2.5 Heavy bed loadi
4.97 0.0683 53.4 43.1 96.5 685 7.80 9.3 4.5 2.3 Pulsating, justabove critical
4.94 0.0680 51.0 41.9 92.9 670 7.60 9.1 4.5 2.3 j CRITICAL I4.81 0.0662 ' 49.8 40.8 c 90.6 660 7.50 9.0 .4.6' 2.3 Deposit, thin bed
4.56 0.0627 42.7 35.1 77 .8 610 6.95 7.6 4.1 2.1 Flat bed
4.00 .·'0.0550 33.6 28.0 61.6 540 6.15 5.6 3.1 1.6 II II'
3.51 0,'0483 27.8 13.80 41.6 440 5.00 4.0 2.3 1.2 II II
I. I-'
WNI
Series BS-03-2
. '{c = 2.30%CRITICAL CONDITION ,
Vc = 7.60 fps
loop readi gtest section". -,..-. n s
ohl. 95 " (..AU...) t.hR A~ AhR+AhD Qm V AhR~AhD AhR-AhD C CONMENTSAt m m
tin.]··-correc ted
[in.] [in.] [in.] [gpm] [fps] [in.) [in.] [%)
-6.56 0.0902 68.0. 51.4 119.4 755 8.30 16.6 10.4 5.3 Everything moving
6.36 0.0874 " 58.7 44.5 103.2 700 7.95 14.2 9.1 4.75 Pulsating, justabove critical
6.36 0.0874 56.8 42.7 99.5 690 7.85 14.1 9.2 4.8 ICRITICALl
6.29 0.0864 53.7 41.3 95.0 675 7;75 12 ..4 7.7 3.9 Deposit
6.13 0.0842 48.3 37.0 85.3 640 7.30 11.3 7.3 3.75 Flat bed
5.44 0.0746 41.6 32.4 74.0 590 6.70 9.2 5.9 3.0 II II"
5.07 0.0697 34.4 26.1 60.5 535 6.10 8.3 5.8 2.9 II II
4.26 0.0585 27.0 . 21. 5 48.5 480 5.45 5.5 3.5 1. 75 II II
,~
"WW,
Series BS-03-3
CRITICAL CONDITION { :
" C
= 4.80%
= 7.85 fps , '~
\,, "! ,t
" '
...... --~.- ---- .
.. "
0.2
150.01 L...-_.&---.L__...L-'---...L-_J..-.J~--L.---L---L_-_....L-
2 3 4 5 6 7 8 9 10
Vm (fps), MIXTURE VELOCITY
0.10.09
CJ) 0.08CJ)
o. 0.07 5%-Ja<tw:c 0.05w Clear Fluida:::::> 0.04l-x~
0.03E
i •.cl~<J<Ji I
0.02
Plot of Series BS-001 Data
-134-
t
loop read'te t r.ection.s ' ci Lngs -~hl.95' (~"!....) 6hR . 6hD AhR+6~ ~ V MlR-AhD AhR-AhD C COMMENTS
A-L mm -orrect:ed
r- in.l [in.] ·[in. ) [in.] [gpm] [fps] [in.] [in.] [%]
'. 2.40 0.0330 53.7 45.7 99.4 700 ,7.9,5 8.0 3.0 1.5 Everything moving-
2.35 0.0324 49.8 42.8' 92.6 675 7.7 7.0 2.5 1.25 Heavy bed load
2.10, 0;0289 46.7 40.0 86.7 650 7.4 6.7 2.4 1.2 " " "1.80 0.0248 43.1 37.1 80.2 625 7.1 6.0 2.1 1.1 Rapidly moving bed
1.85 0.0255 40.1 34.7 . 74.8 600 6.8 5.4· 1.9 1.0 Pulsating bed
1.80 0.0255 36.0 31.2 67.2 570 6.5 4.8 ·L8 0.9 " ",1. 75 0.0241 32.3 28.0 60.3 540 6.15 4.3 1.5 0.75 Slowly pulsating
bed
1. 70 0.0234 29.7 26~2 55.9 515 5.85 3.5 1.2 0.6 ICRITICAL I1.60 0.0220 25.8 23.0 48.8 485 5.5 2.8 0.8 0.4 Thin bed
1.30 0.0179 20.2 18.0 38 ..2 " 420 4.8 1.8 0.4 0.25 Flat bed
0.75 0.0103 11.5 10.6 22.2 315 3.6 0~9 0.1 0.05 " "
,I-'W\.J1
"
Series BS-OOl-l
{
' C = 0.75%CRITICAL CONDITION
V = 5.85 fpsC
! .
/ 1;
../
loop readingste t sections
.t.i\ .95. (-~) 6hR AhD 6hR+6hD ~ ,Vm All -6h . 6hR-6hD C COMMENTS6-L m R D
corrected[in. ] (in. ] '(in. J (in.] (gpm] (fps] [in.] [in.] [%]
2.90 0.0399 61.8 51.2 113.0 730 8.3 . 10.6 4.7 2.4 Everything moving
2.80 0.0386 56.4 46.6 103.0 700 7.95 9.8 4.5 2.3 Heavy bed load
2.70 0.0371 50.0 41.4 91.4 655 7.45 8.6 4.1 2.1 Pulsating, slidin'gbed
2.85 0.0392 47.0 39.0 86.0 645 7.35. 8 ..0 3.8 1.95 Pulsating, slidingbed
..,2.70 0.0371 44.0 36.4 80.4 620' 7.05 .7.6 3.8 1. 95 Just above..
critical,ICRITICAL!
2.80 0.0386 41.4 34.8 76.2 600 6.85 6.6 3.1 1.60 Deposit
2.60 0.0358 35.8 30.7 66.5 570 6.5 5.1 2.1 1.10 Thin bed
2.40 0.0330 31.4 26.6 58.0 530 6.05 4.8 2.3 1.20 Flat bed
2.10 0.0289 26.9 23.0 49.9 485 5.5 3.9 1.9 1.0 " II
1. 75 0.0241 .21. 7 18.8 40.5 430 4.9 2.9 1.4 0.7 II "0.55 0.0076 6.0 5.2 11.2 220 2.5 0.8 0.4 0.2 Dunes
I.....UJ0\I
Series BS-001-2
. CRITICAL CONDITION { CVc
= 1.90%
= 6.95 fps
10 P e ditest section 0 r a ngs
~h1.95· (-Ml..) 6hR 6hn 6hR+6hO Q.. Vm AhR-h~ AhR-AhD C COMMBN'rSA~ m ..\
in.)corrected
[in.] '[in.] [in.] [gpm] [fps] [in.] [in.] [%]
3.20 0.0440 59.9 48.2 108.1 715 8.15 11.7 6.1 3.1 Heavy bed load
3.30 0.0454 53.0 42.8 95.8 670 7.6 10.2 5.5 2.85 Quickly moving, bed
3.40 0.0469 49.7 40.4 90.1 655 7.45 9.3 4.8 2.45 ICRITICALl
3.40 0.0469 42.8 34.4 77 .2 600 6~8 8.4 4.9 2.47 Thin bed
3.20 0.0440 38:0 30.9 68.9 570 6.5 7.1 4.1 2.10 Thickening bed·,
2.90 0.0399 31.1 25.4 56.5 510 5.8 5.7 3.4 1. 75 " " . ..2.40 0.0330 24.85 21.0 45.85 460 5.25 3.85 2.15 1.10 " "1.80 0.0248 19.4 16.5 ·35.9 410 4.65· 2.9 1. 50 . 0.80 Flat bed ,
0.60. 0.0083 6.0 5.3 11.3 220 2.5 0.7 0.30 0.15 Very littlesaltation, dunes
Series BS-001-3
CRITICAL CONDITION {. C. Vc
= 2.50%
= 7.45 fps
.,.,.....
loop ead'ngtest section r ).. s
(..Ah..)\
t\hl. 95· AhR AhO AhR+AhO Qm Vm Aha-Aho AhR-Aho C COMMENT.Sb.L m
corrected[in.) [in.] . ·[in. J [in.] [gpm] [fps] [in.] [in.] [%]
4.10 0.0555 68.10 51.30 119.40 740 8.4 16.80 10.70 5.5 Above critical, .'
slowly pu1sati~g
4.60. 0.0631 64.40 48.20 112.60 720 8.2 16.20 10.50 5.35 Just above'crit ica1 ..
4.70 0.0645 62.10 45.90 108.00 710 8.1 16.20 10.70 5.45 Sliding ..
ICRITICAL I ..
58.40 43.40 101.80 7.85 Just below. .
4.80 0.0660 690 . 15.00 9.90 5.05critical
4.80 0.0660 53.00 39.60 92.60 660 7.5 13.40 8.80 4.50 Flat bed
4.70 0.0645 48.80 36.80 85.60 620 7.05 12.00 8.3 4.2 " "4.40 0.0605 42.80 32.00 74.80 585 6.65 10.80 7.5 3.85 " "
3.90 0.0536 33.40 25.30 58.70 520 5.9 8.10 5.70 2.90 " "3.60 0.0495 28.90 22.25 51.15 480 5.45 6.65 4.65 2.40 " "
2.70 0.0371 21.90 17 .30 39.20 420 4.8 4.60 3.20 1. 65 " "2.00 0.0275 16.50 13.50 30.00 370 4.2 3.00 1. 90 1. 00 Flat bed, little
bed load
0.70 0.0096 5;90 5.10 11.00 215 2.45 0.80 0.40 0.20 Dunes
I....w00I
CRITICAL CONDITION: C = 5.40%V
C= 7.95 fps
Series BS-001-4
. " .~'.
. !
i, - ,'.
:r .·x, '
-"-- .~ -.
0.2
·15
0=6 in.dso=0.45 mm
5=0.027E' =0.000 16 ft.
5 6 7 8 9 10430.01 '---~.I...----I.__...I.-_......L..._.L---L..---I.---L---L -L-
2
U)U)
9o 0.06<Xl&J:I:
.~::> 0.04.....X~
0.03E
:CI~';<J <I,0.02
Vm (fps) ,MIXTURE VELOCITY·
Plot of ~eri~s BS-003 Data
-139-
di1i~test sect on_roop rea ngs ,
6oh1 . 95· (~) .6hR 611' 6hR+6ohD Q. V liliR-6hD liliR-6hD C COMMENTSM, m D m m
corrected[in.] [in. ] ~[in.] [in.] [gpm] [fps] [in.] [in.] [%]
.-- . - . - -_. .-
3.20 0.0440 63.8 55.0 118.8 775 8.84 8.8 . 2.3 1.15 Heavy bed load
2.56 0.0352 49.2 42.5 91.7 675 7.70 6.7 2.1 1.05 " " "2.44· 0.0335 46.2 40.2 86.4 655 7.45 6.0 1.8. 0.9 Sliding bed
2.31 o 0317 43.0 37.5 80.5 625 7.10 5.5 1.7 0.85 Pulsating bed
2.20 ·0.0·304 40.0 35.0 75.0 605, 6.87 5.0 1.5 0.75 " "2.0 0.0274 33.0 28.6 61.6 550 6.25 ·4.4 1.7 0.85 Just above
·critica1
1. 97 . O~ 0271 31.8 27.7 59.5 540 6.15 4.1 1.5 0.75 ICRITICAL I1.77 0.0243 27.6 24.2 51.8 500 5.70 3.4 1.3 0.65 Deposit
1.49 .0.0204 21.4 19.3 40.7 450 5.10 2.1 0.5 0.25 Thin bed
I
I~.p-oI
Series BS-003~1
. . {C· = 0.75%CRITICA.L COND ITION V
c = 6.15 fps
,/
10 P ead'ti.. t,"L.eS sec on 0 r longs"
D.h1 . 95(..M!-.) , AhR
Doh· AhR+AhD ~. V liliR-AhD AhR-AhD C COMMENTSA)" m D m
corrected.. in.] [in.] [in.] [in.] [gpm] [fps] [in.] [in.] [%)
,-- .3.49 0.0480 63.4· 52.6 116.0 .. 745 8.50' 10.8 4.8 2.,4.: Heavy bed load
3.31 0.0455 57.4 48.2 105.6 710 8.10 9.2 3.9 2.0 " " "
3.26 0.0448 53.8 44.4 98.2 670 7.60 9.4 4.5 2.2 Sliding bed
3.31 0.0455 47.4 39.2 86.6 635 7.25 8.2 4.2 2.1 Pulsating bed
[CRITICALI
3.23 0.0444 44.2 37.0 81.0 620 7.05 7.2 3.'5 1.8 Just belowcritical
3.21 0.0441 38.6 32.4 71.0 575 6.55 6.2 3.1 1.6 Thin bed
2.95 0.0405 34.0 28.2 62.2 540 6.15 5.8 3.1 1.6 Flat bed
2.72 0.0374 27.0 23.0 50.0 480 5.45 4.0 2 ..1 1.1 " "2.61 0.0190 19.8 17.2 37.0 410 4.65 2.6 1.2 0.6 " "0.66 0.0092 5.8 5.4 11.2 210 2.40 0.4 0.1 0.05 Dunes
I
~t-'I
Series BS-003-2
{
C = 2.00%
CRITI.CAL CONDIT.ION. V
C--
7.10 fps'"
loop readingstest section,.. ... --t.hl, 95 (..Ah...) , t.hR tJ'lD t.hR+t.~ ~' V tiliR-t.hD tili -t.h C COMMENTS
M~ m m R Dcorrected
[in.) [in. ] [in.] [in.] , [gpm] [fps] [in.] [in.] [%]i
..-,,_.. .4.23 0.058 65.4, 51.0 116.4 730 8.30 14.4 8.7 4.45 Everything moving
I '.1''-
4.51 0~062 57.8 45.3 103.1 685 7.80 ' 12.5 7.6 3.9 Sliding bed
4.56 0.0627 55.3 43.0 98.3 ' 665 7.55 12.3 7.8 4.0 Pulsating bed
4.62 0.0640 55.2 43.5i
98.7 665 7.55 11.7 7.2 " 3.7 Just aboveerit ica1
(ICRITICAL I
4.75 0.0652 53.0 41.8 94.8 655 7.45 11.2 7.0 ' 3.6 Just belowcritical
4.59 0.0630 44.6 35.25 79.85 605 6.90 9.35 5.85' 3.0 F~at bed
4.07 0.0560 34.6 28.30 62.90 540 6.15 6.3 3.7 1.9 " "3.08 ' 0.0422 24.0 .19.60 43.60 ' 450 5.10 4.4 2.7 1.4 " ",
I......poNI
Series BS-003-3
{
C = 3.70%CRITICAL CONDITION ' .
V = 7.50 fpsC
i
, I
"
loop readite t section.. s -, ngs ,~hL 95 . (..A1l..) . ~hR ~h . AhR+AhO ~. V . AhR-AhD AhR-AhD C COMMENTS
A~ m D mcorrected
[in.) [in. ) [in.) [in.) [gpm) [fps] [in.] [in.] [%)
---- '.4.82 .0.0663 69.2· 52.7 121.9 760 8.65 . 16.5 10.3 5.3 Sliding bed
5.10 0.0702 64.0 48.2 112.2 720 8.20 15.8 . 10.3 5.3 II "5.39 . , 0.0741 60.4 45.7· 106.1 700 7.95 14.7 9.6 5.0 Pulsating bed
5.35 0.0738 58.8 44.5 . i03.3 680 7.75 14.3 9.5 4.9 ICRITICALI
5.46 0.0752 55.6 42.6 98.2 670 7.60 13.0 8.5 :l 4.3 Depo.sit
5.46 0.0738 50.2 38.4 '88.6 635 7.25 11.8 7.9 '4.0 Thin bed
5.13 0.0706 44.8 34.0 78.8 600 6.80 10.8 7.4 . 3.~.8 Flat bed
4.82 0.0663 39.5 30.1 69.6 560 6.35 9.4 6.5 3.3 " "'.
4.34 0.0596 34.0 26.2 60.2 525 6.00 7.8 5.4 2.7 " ".3.90 ·0.0536 28.2 22.1 50.3 475 5.40 6.1 4.3 2.2 if "
3.41 0.0470 23.7 18.8 42.5 435 4.95 4.9 3.4 1.7 " II
2.64 0.0364 19.0 15.7 34.7 390 4.45 ' 3.3 2.1 1.1 II II,
I.....~WI
{
c = 5.00%CRITICAL CONDITION
Vc = 7.75 fps
Series BS-003-4
"! "
.".
di1i,. testset~~)n
oop rea ngs
6hH 0 6hR6h
D6h
R+6hD Qm V 6hR-6h
D'·6-t m 6hR-6h
D C COMMENTS. 2 mcorrected
[in. ] [in. ] [in. ] [in. ] [gpm] [fps] [in. ] [in. ] [%]
2.60 0~0183 22.70 21.65 44.35 445 5.05 1.05 0.95 2.10 Total transport,heavy. bed load
2.50 . . 0.0176 . 18.80 17.75 36'.55 400 4.5 1.05 0.95 2.00 Heavy bed load
2.00 0.0141 15.15' 14.30 29.45 355 . 4.05 0.85 0.75 1.70 Thickening bed,
sliding alonginvert
1.90 0.0134 . 13.25 12.40 25.65. 330 3 .. 75 0.85 '. 0.70 1.50 Pulsating bed move-ment with spo-radic settling
1.80 0.0127 12.50 11.70 24.20 320 3.6 0.80 0.65 1.30 Just above critical., condition
1. 70 : 0.0120 11.40 . 10.60 22.00 300 3.4 0.80 0.65 1.25 ICRITICALI1.40 . 0.0099 9.20 9.10 18.80 270 3.05 0~60 0.45 1.00 Sporadic dune be-
havior
I.....t:I
Series BS-PP1-1
1.30%CRITICAL'CONDITION {c ,=
. Vc = 3.40 fps
.' '. ,
di1i,. test .set~~)n oop rea ngs6hR 6h
D6hR+6hD Qm V6hHO ' M" m 6hR-6h
D6hR-6hD C . COMMENTS2' m
corrected[in. J [in. J "[ in.] [in. ] [gpm] [fpsJ [in. J [in. J [%]
3.35 0.0236 28.20 26.80 55.00 500 5.7 1.40 1.25 2.80 Full bed loadtransport
3.00 . 0.0212 24.00 22.70 46.70 460 5.25 1.30 1.15 2.50 Slowly moving bedload ~
3.05 0.0216 20.30 19.15 39.45 420 4.8 1.15 1.05 2.35 Sliding bed
2.90 0.0204 18.55 17.45 36.00 395 4.55 1.10 1.00 2.20 " ".
2.80 0.0197 { 16.65 15.75 32.40 365 . 4~ 15 0.90 0.85 1.95} Sporadic',15.40 14.40 .29.80 360 4.1 1.00 0.90 2.00 pulsating trans-
port conditions2.70 0.0,190 14.05 13.10 27.15 . 345 3.9 0.95 0.85 1.95 Just above cri~.
condition
2.50 0.0176 13.45 .12.60 26.05 340 3.85 0.85 0.75 1.70. ICRITICAL I2.10 0.0480 11.45 10.75 .22.20 315 3.6 0'.70 0.60 1.30 Infrequent. duning
concentrations
I.....J:'oVII
.. Series BS-PPl-2
1.90%C~ITICAL CONDITION {c I =. Vc = 3.85 fps
/)
di1tit t,. esset~~)n
oop rea ngs
f1hH 0 ~hR f1hD f1hR+~hD Qm V
f1hR-~hD ~~-M1D C COMMENTS2 ~t m mcorrected
[in. ] [in. ] [in.] [in. ] [gpm] [fps] [in. ] [in. ] . [%]
,,
3.90 '0.0275 25 •.80 24.15 49.95 485 5.5 1.65 1.59 3'.30 Total transportbasically bedload
3.95 0.0279 23.20 21.60 44.80 455' 5.15 1.60 1.45 3.20 He~vy bed load
4.10 0.0289 21.10 19.60 40.70 440 5.00 1.50, 1.40 3.15 Sliding, thicken-, ing bed load
4.10 0.02"89 18.95 17.55 36.50 ' 415 4.75 1.40 ' , 1.30 3.00 Pulsating just, ,
above crit.condition of
"
, , bed stoppage
3.80 ' 0.0268 17.70 ' 16.20 33.90 395 4.55 1.40 1.30 3.00 Almost crit.
~,[CRITICAL I
4.10 0.0289 1l~.45 13.30 27.75 ',360 4.10 1015 1. U5 , 2.3 Bed, and long dune
, . build-up
.,
Series BS-PPl-3
..
CRITICAL CONDITION {CVc
=-3.00%
= 4.45 fps
':.'
','.' '
.' ",~ " .... ", t •
""j '.1 , ,i
J]
di1tit t~ esset~~)n
oop rea ngs
6hH 0 6hR6h
D6hR+6hD Qm V
~hR-6hD ~hR-~hD C COMMENTS2 M"m mcorrected
[in. ] [in. ] [in.] [in. J. [gpm] [fps] [in. ] [in. ] . [%]
4.70 0.0332 29.00 26.55 55.55 515 5.9 2.45 2,30 5,.10 Most 'all transport',in form ,of heavybed load
4.70 '0.0332 27.25 24.95 52.20 495, 5.6 2.30 2.10 4.70 Bed load
4.50 0.0318 26.25 23.95 50.20 485 ' 5~5 2.30 2.10 ' 4.65 Slow movingthickening bed
4.80 ' 0.0338 24.30 22.05 46.35 , 470 5.35 2.25 ' ' 2.05 ' 4.55 Effective scourmechanism
4.90 0.0345 23,.05 21.00 44.05 , 450 5. i.o 2.05 1.95 4.30 Pulsating bed
4.55 .. 0.0320 19.05 17.15 36.20 415 4.75 1.80 1.70 3.90 Slugs of varyingconcentration
~ ICRITICAL f3.90 . 0.0275 16~30 14.60 30.90 ' 380 4.35 r.70 1.60, 3.50 Deposit condition,
.. impulsive.. dune, " , motion
•' ,
, .
Series BS-PPl-4
'.
{
c =3 807.:CRITICAL CONDITION " 0
, ' VG
== 4.60 fps
/
APPENDIX C: REGRESS.ION ANALYSIS DATA
A regression analysis was made to· correlate each of three-modifiedFroude numbers (I), (II), and (III), as defined in Section 4.1 of thecontents, with the following parameters: concentration C; concentration;C, and particle diameter, d; and concentration, C, and relative particlesize, diD. The results of this analysis are tabulated in Tables C.l(a),C.l (b), and C. 1 (c) for each Froude number.
The modified Froude numbers were calculated with solids concentration, C,over five different ranges of data. Correlation was alsoevaluated for regression of each modified Froude number with both solidsconcentration, C, and either particle diameter, d, or relative particlesize, diD, over tWo ranges of data~ These ranges are specified inTables C.l(a), C.l(b), and C.l(c) along with indications of "goodnessof fit".
The regression analysis fits data to a geometric curve, correlatinglogarithmic values on a linear or arithmetic scale, as given with:
1\I
Log Fr = ka Log C + Log k1
Reconverting to arithmetic scale gives the form:
(C. 1)
(4.1)
Likewise, for a multiple regression analysis with modified Froude number,F , solids concentration, C, and either particle diameter, d, or relativepirticle size, diD, the linear form for log-log data fitting is givenw~h: -
Log Fr = k4 Log C + ~ Log d + ~og k3
and subsequently written as:
(C.2)
(4.2a)
Standard deviation, S.D., coefficient of correlation, R, and standarderror of estimate, S , are given for each analysis listed in Table C.2,defined respectivelyYas:
-148-
/
Vc f (C)=J 2gD (55-1)2
Range No. of Equation S.D. R SData y
d = 0.88 mm 22 F = 0.901 CO.O S6 0.049 0.845 0.0264r
d = 0.45 ImIl 24 . F = 0.892 CO.13l 0.088 0.935 0.0311r
d = 3.63 mm .4 F = 0.~09 CO.29O 0.052 0.994 0.0059r
d = 0.45 to 46 F = 0.893 CO.1l4 0.073 0.886 0.03360.88 mm r
all d 50 F = 0.905 CO.122 0.078 0.872 0.0380r
Vc f (C,d)=.j2gD (5 -1) 2
5
RangeNo. of
Equation RData
d = 0.45 to 46 F = 0.921 CO.109 dO.05S 0.8710.88 mm r
all d 50 F = 0.927 CO. l10 dO.07O 0 0 863r
Vc d= f2 (C'u)V2gd •(5 -1)5
No. ofRange Data Equation ·R
d = 0.45 to 0.113 dO.O O246 F = 0.905 C 0.8790.88 ImIl r D
0.114 .d 0.002all d 50 F = 0.905 C 0.818r D
Table C.1(a): Correlation with Modified Froude Number (I)
-149-
/
Vc - [1 - tan eJ = fa (C)V2gD (s -I)'
s
Range No. of Equation S.D. R SData y
d = 0.88 mm 22 F = 0.908 CO-08O 0.047 0.831 0.0259.r
ci = 0.45 mm 24 F = 0.900 CO-la4 0.084 0.919 0.0332r
d = 3.63 mm 4 F = 0.909 Co-a9o 0.052 0.994 0.0059r
d = 0.45 to 46 F = 0.901 CO-106 0.069 0.870 0.03430.. 88 mm r
all d 50 F = 0.912 CO. 114 0.075 0.854 0.0387r
Vc [1 - tan eJ = f (C,d)"/2gD
\ a• 0, (s -1)s
RangeNo. of Equation RData
d = 0.45 to 46 F = 0.928 CO-lOS dO.OS6 0.8770.88 mm r
- all d 50 F = 0.934 CO. 106 dO.068 0.866r
Vc[1 - tan eJ fa
d= (C'i))
" 2gD(s -15
s
RangeNo. of Equation RData
d = 0.45 to 0.108 d 0.00246 F = 0.913 C 0.8840.88 mm r· D
0.110 d 0.002all d 50 F = 0.912 C 0.820r D-
Table C.2(b): Correlation with Modified Froude Number (II)
-150-
./
/Vc-;:::===============. = f 2 (C)
"2gD (ss-l) [1 + tan eJ'
Range
d = 0.88 mm
d = 0~45 mm
d = 3.63 mm
d = 0.45 to0.88 mm
all d
No. ofData
22
24
4
46
50
Equation·
Fr
= 0.905 CO.0 82
F = 0.896 CO.127
r
F = 0.909 CO.290
r
F = 0.898 Co.nor
F = 0.909 CO. U8
r
S.D.
0.048
0.086
0.052
0.071
0.076
R
0.839
0.928
0.994
0.880
0.864
SY
0.0259
0.0319
0.0059
0.0336
0.0381
Vc-=======;=:====:::; = f (C,d)j2gD (ss-l) [1 + tan eJ' '"2
I •
Range
d = 0.45 to0.88 mm
all d
No. ofData
46
50
Equation
F = 0.925 CO.107 dO.0 57
r
R
0.878
0.867
vC d
-;:::=======::;::======~ = f.., (C, -D )J2gD (s s-l) [1 + tan eJ' ""
Range
d = 0.45 to0.88 mm
all d
No. ofData
46
50
Equation
0.110
F = 0.909 Cr
0.112
F· = 0.909 Cr
d 0.002
D
d 0.002
D
0.885
0.822
Table C.3(c): Correlation with Modified Froude Number (III)
-151-
/
or:
R
explained variationtotal variation
2
. I: (F t - F)es
-2I: (F - F)
(C.3)
(C.4)
-2where I: (Fest - F) is the sum of the deviati2.n~ of fitted (or estimated)values from the average, squared; and I: (F - F) is the sum of the deviations of actual data values from the average, squared.
where N is the total number of data analyzed.
(C.6)
I .
'!he standard error of estimate, S , includes both central tendency, related .to standard deviation, s.n.; and variability, described by thecoefficient of corre·lation, R, in indicating "goodness of fit".
One is warned that the coefficient of correlation, R, determinedon a log-log scale, as reported in this study, may give a misleadingindication of "goodness of fit" that would be found on an arithmeticscale. Log-log data near to the origin have the strongest influenceon the regression. Since most of the Lehigh.data were ohtained at lowsolids concentrations, 0.10 < C< 2.0%, log-log fitting works to ouradvantage. Correlation, on the other hand; weighs everydat·a pointequally, and an insignificant change in regression at a high solidsconcentration data point may mistakenly infer greatly improved correlation, or vica versa. For a closer look at the·raw data which deter~·:
mined best-fit, the regression analysis data output is on file inFritz Laboratory at Lehigh University.
Some resulting best-fit equations, from the Froude number (II)analysis, are presented in Figs. C.1 to C.3, inclusively. Figure C.lshows the best-fit equation for modified Froude number, Fr ,. correlated with solids concentration, C, as evaluated for each of the threetested particle diameters, d. A relationship between sand arid plasticpellet results is not immediately recognized. However, the similarityin form exhibited between the equations for sand is to be expected,subsequent to a study of Gibert (1960). .
-152-
.,'
0.290'-----+ Fr = 0.909 C
. (d =3.63 mmJ
1 !L-'T:::::=±==:JC8==G~=.;;;;;;;,;;;;a-;I-~VI~~:::~~--
0.124Fr = 0.900 C· .
(d =0.45 mm.)
o.oao~-Fr =0.90aC
(d=0.88 mm,)
·vC
]..001----=---
1.a6
..
.....VIW
•
C[%]
o .. 1.0 3.0 4.0 6.0 6.0
I . . Fig. C.l: Equations Best Fitting Modified Froude Number with Solids Concentration;Particle Diameter (for two different sands and plastic pellets) as Parameter
0.106 0.068Fr =0.934.C d
'-
___--------------r d=3.63----
1.00I-__---,,;::;:---_~
1.26
• •....VI.pI
O'601:=-- ~----:._1_--
•c[%]
o 1.0. 3.0 4.0 6.0 6.0
. Fig. C.2:Equation . Best Fitting Modified .Froude Number with Solids Concentration and ParticleDiameter, Evaluated for Both Sand-Water and Plas~ic Pellet-Water Data
0.110 0.002F: ': 0 912 C diD. r ·
Vc .-;:::===~. [1 • tan 8J/2gD (ss·l)
o ,'~ B
1-----:0=-----9G-~JB"'"~~===::;::-=J:==-=-~~~t):=-:=-=~
'A~
o..l'~-4---:e
1.a6
t....U1U1t
C[%]
o 1.0 , 2.0 3.0 4.0 6.0 6.0
Fig. C.3: Equation Best Fitting Modified Froude Number with Solids Concentration and RelativeParticle Size, Evaluated for Both Sand-Water and Plastic Pellet-Water Data
/
Figure C.2 illustrates the effect of including particle diameter,d, as an independent variable in correlating all of the data. Sincethere are relatively few data points for sufficiently expressing thetrend of the plastic pellets data, the sand particles dictate the general form of the function. However, it should be noted that the plasticpellets significantly impinge upon the form of the sand particle curvesat low concentrat.ions •... It is to this end that use of Fig. 2.3 and theassociated relationship is discouraged.
Figure C.3 gives the relationship for Froude number (II) fittedwith solids concentration, C, and relative particle s-ize, diD, overthe entire range of data. The inclusion of diD is relatively neg1i- .gib1e, and the effect due to different particle diameters, d, is essentially eliminated. Further, the plastic pellet data impose a greaterrelative influence on the regression than indicated in other correlations of the total data. -The relationship given with Fig. C.3 is alsonot recommended.
- -.
-156-
REFERENCES
1. Barr, D. and Ridell, J. (1968): "Homogeneous Suspensions in Circular Conduits: A Discussion"; Proc. Amer. Soc. of Civ. Engrs.,Vol. 94,. PLl.
2. Blatch, N. S. (1906): '~orks for the Purification of the WaterSupply of Washington: A Discussion"; Trans. Amer. Soc. of eiv.Engrs., Vol. 57.
3. Bonning.ton, S. T. (1961): "Estimation of Pipe Friction Involvedin Pumping Solid Material"; Brit. Hydromechanics Res. Assoc.,TN 708, December.
4. Bruce, W. A., Hodgson, G. W., and Clark, K. A. (1952): "HydraulicTransportation of Oil-Sand Tailings in Small-Diameter Pipes";Trans. Can. Instn.of Mining and Met., Vol. 15, December.
5. Carstens, M. R. (1969): "A Theory for Heterogeneous Flow ofSolids in Pipes"; Proc. Amer. Soc. of Civ. Engrs., January,HYl.
6. Carstens, M. R. (1971): Discussion of "Sediment TransportationMechanics: J. Transportation of Sediment in Pipes", byH. W. Shen et al., July 1970; January, HY1.
7. Condolios, E. and Chapus, E. E. (1963a): "Transportating SolidMaterials· in Pipelines"; Chern. Engrg., June-July, p. 93.
8. Condolios, E. and Chapus, E. E. (1963b): "Designing SolidsHandling Pipelines"; Chem. Engrg., June-July, p. 131.
9. Condolios, E. and Chapus, E. E. (1963c): "Operating Solids Pipelines"; Chern. Engrg., June-July, p. 145.
10•. Charles, M. E. (1970): "Transport of Solids by Pipeline"; Proc.of the First Intern. Conf. on the Hydraulic Transport of Solidsin Pipes, Coventry, England, Brit. Hydrodynamics Res. Assoc.,September.
11. Durand, R. (1953): "Basic Relationships of the Transportation ofSolids in Pipes - Experimental Research"; Proc. 5th Congo ofIntern. Assoc. Hydr. Res., Minneapolis.
12. Durand, R. and Condolios, 1;:. (1956): "Technical Data on HydraulicTransport of Solid Materials in Conduits"; Revue de L'IndustrieMinera1e, Numero Special IF, June.
13. Einstein, H. A. and W. H. Graf (1966): "Loop System for MeasuringSand-Water Mixtures"; Proc. Amer. Soc. of Civ. Engrs., January,HYL
-157-
14. Ellis, H. S., Redberger, P. J., and Bolt, L. H. (1963a): "Transporting Solids by Pipe Line: Basic Principles and Power Requirements"; Ind. and Engrg. Chem., Vol. 55, No.8, August.
15. Ellis, H. S. and Round, G. F. (1963b): "Laboratory Studies on theNickel-Water Suspensions"; Can. Mining and Met. Bulletin, October.
,I
" " ""16. Fuhrboter, A. (1961): "Uber die Forderung von Sand-Wasser-Gemischen·in Rohr1eitungen"; Mitt. d. Franziu~-Inst., Techn.,Hochschu1e, Hannover, Heft 19.
17. Gibert;::R. (1960): "Transport Hydraulique et Refou1ement desMixtures en Conduit"; Anna1es des Pontes et Chaussees, l30eAnnee, No. 12, et No. 17.
18. Govier, G. W. and Charles, M. E. (1961): "The Hydraulics of thePipeline Flow of Solid-Liquid Mixtures"; The Engrg. Journal,August.
19. Graf, W. H. and Acaroglu, E. R. (1966): "Settling Velocities ofNatural Grains"; Intern. Assoc. of Sci. Hydrology, Xre Ann~e,No.4, December.
20. Graf, W. H. and Acaroglu, E. R. (1968): "Sediment Transport inConveyance Systems (Part 1): A Physical Model for SedimentTransport in Conveyance Systems"; Intern. Assoc. of Sci.Hydrology; XIIre Ann~e, No.2.
" "21. Graf, W. H., Robinson, Jr., M. P., and. Yucel,· O. (1970): "TheTransport of Solid Suspensions in Conduits, Part I: The Critical Velocity for Solid-Liquid Mixtures"; Fritz Engrg. Lab.Report No. 353.1, Lehigh Univ., Bethlehem, Penna.
22. Graf, W. H. (1971): "Sediment Transport in Closed Pipes"; Hydraulics of Sediment Transport, McGraw-Hill Publishing Co.,Inc., New York, New York.
23~ Howard, G. W. (1939): "Transportation of Sand and Gravel in aFour-Inch Pipe"; Trans. Amer. Soc. of Civ. Engrs., Vol. 104.
24. Hunt, W. A. and Hoffman, I. C. (1968): "Optimization of PipelinesTransporting Solids"; Proc. Amer. Soc. of Civ. Engrs., October,PLl.
25. Lowenstein, J. G. (1959): "Design so Solids Can't Settle Out";Chem. Engrg., January.
26. Newitt, D. M., Richardson, J. F., Abbott, M., and Turtle, R. B.(1955): "Hydraulic Conveying of Solids in Horizontal Pipes";Trans. Inst. Chem. Engrs., Vol. 33/2.
-158-
27. O'Brien, M. P. and Folsom, R. G. (1937): "The Transportation ofSand in Pipe Lines"; Univ. of Calif. Publications in Engrg.,Vol. 3.
28. Roberts, R. N. (1967): "Pipelines for Process Slurries"; Chem.Engrg., July.
" "29. Robi,nson, Jr., M. P., Yuce1, 0., and Graf, W. H. (1970): "TheTransport of Solid Suspensions in Conduits, Part II: ModifiedVenturimeter; A Measuring Device for Solid-Liquid Mixtures";Fritz Engrg. Lab. Report No. 353.2, Lehigh Univ., Bethlehem,Penna.
" "30. Robinson, Jr., M. P. and Yuce1, o. (1971): "Discussion of Sedi-ment Transportation Mechanics: J.Transportation of Sedimentin Pipes", by H. W. Shen et a1., July 1970; March, HY3. .
31. Rose, H. E. and Duckworth, R. A. (1969): "Transport of SolidParticles in Liquids and Gases"; The Engineer, March.
32. Shen, H. W., Karaki, S., Chamberlain, A. R., and Albertson, M. L.(1970a): "Sediment Transportation Mechanics: .J•. Transportationof Sediment in Pipes"; Proc. Amer. Soc. of Civ. Engrs., July,HY7.
33. Shen, .H. W. and Wang. W. S. (1970b): "Incipient Motion and Limit-·ing Deposit Conditions of Solid-Liquid Pipe Flow"; Proc. of theFirst Intern. Con£. on the Hydraulic Transport of Solids inPipes, Brit~ Hydromechanics Assoc., Coventry, England, September.
34. Sinclair, C. G. (1962): "The Limit Deposit-Velocity of Heterogeneous Suspensions"; Proc. Symp. on the Interaction BetweenFluids and Particles, Inst. of Chem. Engrs.
35. Smith, R. A. (1955): "Experiments on the Flow of Sand-Water Slurries in Horizontal Pipes"; Trans. Instn. Chem. Engrs., Vol. 33.
36•. Spells, K. E. (1955): "Correlations for Use in Transport ofAqueous Suspensions of Fine Solids Through Pipes"; Trans.Instn. Chem. Engrs., Vol. 33.
37. Thomas, D. G. (1962): "Transport Characteristics of Suspensions:Part VI, Minimum Transport Velocity in Large Particle SizeSuspensions in Round Horizontal Pipes"; Amer. Ins.t. of Chem.·Engrs. Journal, July. .
38. Wasp. E. J., Aude, T. C., Thompson, T. L., and Bailey, C. D.(1967): "Economics of Chip Pipe lining" ; TAPPI, Vol. 50,No.7, July.
-159-
39. Wasp, E. J., Aude, T. C., and Kenny, J. P. (1970): "DepositionVelocities, Transition Velocities, and Spatial Distribution ofSolids in Slurry Pipelines"; Proc. of the First Intern. Conf.on the, Hydraulic Transport of Solids in Pipes, Brit. Hydromechanics Assoc., Coventry, England, September.
40. Wilson, K. C. (1965): "Derivation of the Regime Equations forPressurized Flow••• "; Civ. Engrg. Dept., Queen's Univ., Kingston(Ontar io), Report No. 51.
41. Wilson, W. E. (1942): '~echanics of Flow with Non-Colloidal InertSolids"; Trans. Amer. Soc. of C~v. Engrs., Vol. 107.
42. Wood, D. J. (1966): "An Explicit Friction Factor Relationship";Civil Engineering, Amer. Soc. of Civ. Engrs., December, p. 60.
43. Worster, R. C. and Denny, D. F. (1955): "Hydraulic Transport ofSolid Materials in Pipes"; Proc. Inst. of Mech. Engrs.,Vol. 169/32 •
. '.
-160-
/
VITA
The author, Millard P. Robinson, Jr., was born to
Millard P. Robinson and Louise C. Robinson on November, 24, 1947.
At that time he was the first and to be the only brother of a ten
year old sister, Joanne.
In 1959, the author resided in Baumholder, Germany for
one year with his family and returned the next year to enroll at
Springfield High School in Springfield, Pennsylvania. Upon gradu
ation from Springfield in 1965, the author was enrolled as a Civil
Engineering undergraduate at Lehigh University in Bethlehem,
Pennsylvania. His academic endeavors continued in 1969 as a can
didate for the Master'~ Degree in Civil Engineering, with a major
interest in Hydraulics and Sanitary Engineering.
The author's engineering experience has been limited to
summer work in water systems and environmental engineering, but he
bas maintained an active status as an associate member of ASCE and
author of three technical publications. Employment within the En
vironmental Division of Gilbert Associates, Inc., located in Reading,
Pennsylvania, will follow receipt of the MSCE.
The author's fiancee, Mirei11e A. Haudricourt,wi11 become
Mireil1e H. Robinson on June 26, 1971.
-161-
CASE NO.1 F~OUDE NO. (1)TH B ST-FIT E U ON IS •••
LO
EQUATION ON ARITHMETIC SC~LE IS •••
• 0660F = .9Ul0 C
.12 .65561
.1 • •
.20 .76456• • •.50 .64052 .64887.60 ·. 0 •
1.00 1.07587 .90098.00 • • 91
1.75 .96660 .94538• 0 • •
5.00 1.00022 1.03465• 0 • •
1.00 .90098• 0 • •
7.00 1.06501• • .8
1.10 .908393.00 • •5.00 1.01570 1.034651.00 .8 8 •2.3 1.04315 .96785
-{
CA SE NO.2 FROUDE NO. (1)
THE BEST-FIT EQUATION IS •••
LOGF = ( .1309). LOGC + ( -.0497)
EQUATION ON ARITHMETIC SCALE IS •••
I .1309 I----------:--:F=--=--.-:a=-=9::-:1,.-;:9::----=C::-----,I----.----
STANGARD DEVIATION = .0878
CORRELATION COEFFICIENT = .9350
STANDARD ERROR OF ESTIMATE = .n311
c(PERCENT)
.05
.10
.20
.30
.651.001.201.503.007.00
.05
.10
.25
.552.252.50
.751.902.505.40
.752.003.705.00
FROUDE NO.(01 MENSI 0 NLESS)
.46229
.68923
.80690
.91517
.85733
.95819
.98341
.941381.050651.09266
.62198
.65561
.75647
.85733
.92457
.95819
.80295
.953931.022561.09119
.84413
.974521.029421.06374
FITTEDFROUDE NO.
(DIMENSIONLESS)
.60258
.65980
.72246
.76184
.84297
.89187
.91341
.940481.029791.15057
.6D258
.65980
.74387
.82474
.991741.00551
.65691
.9111031.005511.11214
.85891
.976561.058451.10100
CASE NO.3 FROUOE NO. (1)------~T'PTH-noE..--,..,B.......ES~TTtalT~Tr01rr- ....S-.-••-----------------
----------1 F = .9086 C .289811 _
STANDARD OEvr-ATlON = .0524
CORRELATION COEFFICIENT = .9931
STANDARD ERROR OF ESTIMATE - .OOSq
c(PERcENT)
FROUDE NO.(OIHENSI 0Nt Ess)
F-ITTFOFROUDE NO.
eO UfE"ffS I ONLEW
5"S'"Tj------
1.30 .97244 .98036---------------,1-.---=9-=0--------,-1-.-'-'1O~1,-,1.-:4-----~1~.---..->O 943.-4--------
3.00 1.27215 1.249253.80 1.31565 1.33784
CASE NO.I+ FROUDE NO. (1)
THE BEST-FIT EQUATION IS •••
LOGF = ( .1135)· LOGe + ( -.01+90)
EQUATION ON ARITHMETIC SCALE
•11351
IS •••
STANDARD DEVIATION = .0726
CORRELATION COEFFICIENT = .8863
STANDARD ERROR OF ESTIMATE - .0336
C(PERCENT>
.12
.15
.20
.50
.50
.601.001.001.752.005.00
.501.003.007.00
.801.103.005.001.002.30It.80
.05
.10
.20
F~OUDE NO.(DIMENSIONLESS)
.65561
.78168
.85733.89936.8lt052.97500
1.07587.921+57.96660.96660
1.00022.80690.85733.89936.8lt052.878lt4.91962.99511
1.01570• 878ltlt
1.0lt3151.077lt6
.lt6229.68923.80690
FITTEDFROUQE NO.
(DIMENSIONLESS)
.70235
.72036
.71+lt26
.82580
.82580
.81+306
.89337
.89337
.95193
.966lt61.07234
.82580
.893371.011961.11407
.87103
.903081.011961.0723lt
.89337
.961911.06739
• 6359lt.68797.74lt26
CASE NO.4 FROUDE NO. (1)
C(PERCENT>
FROODE Kia.(DIMENSIONLESS)
FITTEDFROOoE NO.
(DIMENSIONLESS)
.30 • 91617 • 7T9~1l
.65 .85733 .850751. 0 0 • 951rr.....9.....-------.8"'9"3"37--------1.20 .98341 .912041. 50 • 94138 • 9-:3-s-z.~'---------3.00 1.05065 1.nli96
------------....7-......0.....0------,..1--...O-g-ZO 8 1. rfli07--------.05 .62198 .63594.10 .65561 .68791.25 .75647 .763~4.55 .8573"3 .831+7''''8--------
2.25 .92457 .979462.50 .95819 .99124
.75 .80295 .864681.90 .95393 .960B52.50 1.02256 .991245.40 1.09119 1.08175--~-----
.75 .84413 .86~68--:-------------..2-...0....0--------......97·45~ .96l"')4cz6,----------
3.70 1.02942 1.036335.00 1.06374 1.07234
CASE NO.5 FROUOE NO. (1)
THE BEST-FIT EQUATION IS •••
LOGF = ( .1218)· LOGC + ( -.0431+)
EQUATION ON ARITHMETIC SCALE IS •••
-----------1 F = .9049 G .1aoll-- -STANDARD DEVIATION = .0777
CORRELATION COEFFICIENT = .8720
STANDARD ERROR OF ESTIMATE - .0380
C(PERCENT>
.12
.15
.20
.50
.50
.601.001.001.752.005.00
.501.003.007.00
.601.103.005.001.002.304.80
.05
.10
.20
FROUOE NO.(01 MENSI ONLESS)
.65561
.78168
.85733
.89936
.84052.• 975001.07587
.92457
.96660
.966601.00022
.80690
.85733
.89936
.84052
.87844
.91962
.995111.01570
.678441.043151.07746
.46229
.68923
.80690
FI1Tf.:DFROUDE NO.
(DIMENSIONLESS)
.69699
.71824
.7ft385
.83165
.83165
.85032
.90489
.90469
.96871
.984591.10080
.83165
.904891.034421.14684
.88064
.915461.034421.10081]
.904891.001491.09535
.62831
.68364
.74385
CASE NO.5 F~OUOE NO. (1)
FITTED-------------....C------F.....R....O....O....OE...-.N....O-.----F....~.....OarrEl\Jrno-.--------
(P ERCENT) (01 MENSI ONL ESS) (D1MENS IONLESS)
.30 • 910"T7 • 78T5lJ
.65 .85733 .85665------------"TI-.'WO'1'1'"0-------."'RqOW-S"'"A2"'i""9·------;0.9'U1t"'""8'1"'W"1"1'y---------
1.20 .963~1 .92521-------------...,..1-.-..-5..0--------.-=9.....~ .....1=3..8------....95lJ6--9,..----------
3.00 1.05065 1.034~2
7.00 1.09268 1.1~68~----------------
.05 .62198 .62831
.io .6~561 .68364
.25 .756~7 .76~34
.55 .85733 .641362.25 .92457 .99681
-------------=2-.-..-5..0-------.-=9---5..,.-8.....1..9-------,-1--.0.01.TT1.-----------------.75 .80295 .87374
1.90 .95393 .978462.50 1.02256 1.D11715.40 1.09119 1.11117
.75 .64413 .87374--------------:2-.-=-0=-0--------.-=9=7.,--4=-5="2-------.~9845...9---------
3.70 1.02942 1.061175.00 i.06374 i.iom601.30 .97244 .93~27
1. 90 1 • 1 0114 • 91 6463.00 1.27275 1.034423.80 1.31565 1.06~63
CASE NO.4TOTAL NUMBER OF DATA = 46
CORRELATION COEFFICIENT = .871
SOLUTION VECTOR IS •••
.1088 .0588 -.0357
EQUATION ON ARITHMETIC SCALE IS •••
IF I.1088 .0588= .9211 C 050
-
FITTEDFROUDE FROUDE
C 050 NUMBER NUMBER{I> (!)
{PERCENT> (MM)
.12 • 86 .65561 .72566
.15 .88 .78168 .74371
.20 .88 .65733 .76736
.50 .88 .69936 .64782
.50 .86 .64052 .84782
.60 .86 .97500· .864811.00 .88 1.07587 .914251.00 .88 .92457 .914251.75 .88 .96660 .971652.00 .88 .96660 .985685.00 .88 1.00022 1.08925
.50 .88 .60590 .847821.00 .88 .85733 .914253.00 .88 .89936 1.030357.00 .68 .84052 1.12988
.80 .88 .87844 .892311.10 .88 .91962 .923783.00 .88 .99511 1.030355.00 .88 1.01570 1.069251.00 .68 .87844 .914252.30 .88 1.04315 1.000994.80 .88 1.07746 1.08442
.05 .45 .46229 .63437
.10 .45 .66923 .68408
CASE NJ.5TOTAL NUMBER OF DATA: 50
CORRELATION COEFFICIENT = .863
SOLUTION VECTOR IS •••
.1097 .0704 -.0329
EQUATION ON ARITHMETIC SCALE IS •••
I '0:1.1097F = .9270 C 050
FITTEDFROUOE FROUDE
C 050 NUMBER NUMBER(!) (D
(PERCENT) ( MM)
.12 .88 .65561 .72811
.15 .88 .78168 .74614
.20 .88 .85733 .770 06
.50 .88 .89936 .85145
.50 .88 .84052 .85145
.60 .88 .137500 .868641.00 .68 1.07587 .9186g1.00 .88 .92457 .918691.75 .88 .96660 .976832.00 .88 .96660 .991245.00 .88 1.00022 1.09601
.50 .88 .80690 .851451.00 .88 .85733 .918693.00 .88 .89936 1.036317. 00 .88 .84052 1.13721
.80 .88 .87844 .896481.10 .88 .91962 .928343.00 .88 .99511 1.036315.00 .88 1.01570 1.096011.00 .88 .81844 .918692.30 .88 1.04315 1.006554.80 .88 1.07746 1.09112
.05 .45 .46229 .63094
.10 .45 .68923 .68077
CASE NO.5TOTAL NUMqER OF DATA = 50
FITTEDFROUDE fRoOtrf:
C 050 NUMBER NUMBER<T) (I)
(PERCENT> ( MM)
• 20 .45 .80690 • 73'7i"5'3.30 .45 .91617 .76792.65 .45 .857n • ~S-87
1.00 .45 .95819 .876301.20 .45 • g-aj4f .8""91+0-01.50 .45 .94138 .916153. 00 .45 1.05065 • 986497.00 .45 1.09268 1.08474
.05 .45 .62198 .5~094
.10 .45 .65561 .68077
.25 .45 .75547 .7"5-2-T2
.55 .45 .85733 .820702.25 .45 .92457 • 957802.50 .45 .95819 .96893
.75 .45 .80295 .8-zt9lf91.90 .45 .95393 .940202.50 .45 1. 02ff6 .9089~
5.40 .45 1.091113 1.05431• 75 .45 .84413 • 81+909
2.00 .45 .137452 .1345513.70 .45 1.02942 1.0111+95.00 .45 1.06374 1.045451.30 3.63 .97244 I. 044141.90 3.63 1.11H14 1.089133.00 3.63 1.27275 1.145073.80 3.63 1.31565 1.17511+
CASE NJ.4TOTAL NUMBER OF DATA = 46
CORRELATION COEFFICIENT = .879
SOLUTION VECTOR IS •••
.1126 .0021 -.0434
EQUATION ON ARITHMETIC SCALE IS •••
I F
.1126•
00211= .9048 C (50/1)
FITTEDFROUOE FROUOE
C 050/0 NUMBER NUM8ER(1) (n
(PERCENT)
.12 .00866142 .6556 .7055
.15 .00866142 .7817 .7235
.20 .00866142 .8573 .7473
.50 .00866142 .8994 .8285
.50 .00866142 .8405 .8285
.60 .00866142 .9750 .84571.00 .00866142 1.0759 .89571.00 .00866142 .9246 .89571.75 .00866142 .9666 .95402.00 .00866142 .9666 .96845.00 .00866142 1.0002 1.0737
.50 .00866142 .8069 .62851.00 .00866142 .8573 .89573.00 .00866142 .8994 1.01377.00 .00866142 .8405 1.1151
.80 .00577428 .6784 .87281.10 .00577428 .9196 .90463.00 .00577428 .90,51 1.01285.00 .00577426 1.0157 1.01271.00 .00577428 .8784 .89502.30 .00577428 1.0431 .98304.80 .00577428 1.0775 1.0678
.05 .00442913 .4623 .6384
.10 .00442913 .6892 .6902
CASE NO.!JTOTAL NUMBER OF DATA = 4(p
FITTEDFRoUDE FROODE
C 050/0 NUMBER NUMBERen rn
(PERCENT>
.20 .00442913 .8069 .7462
.30 .00442913 .9162 .7811
.65 .00442913 .6573 .85211.00 .00442913 .9582 .89451.20 .00442913 .9834 .91301.50 .00442913 .9414 .93623.00 .00442913 1.0507 1.01227.00 .00442913 1.0927 1.1135
.05 .00442913 .6220 .6384
.10 .00442913 .6556 .6902
.25 .00442913 .7505 • 76s-2
.55 .00442913 .8573 .83622.25 .00442913 .9246 .980U2.50 .00442913 .9582 .9917
.75 .00295276 .8029 .86-;-21.90 .00295276 .9539 .96072.50 .00295276 1.0226 .99085.40 .00295276 1.0912 1.0805
.75 .00295276 .8441 .86522.00 .00295276 .9745 .96623.70 .00295276 1.0294 1.03555.00 .00295276 1.0637 1.0712
CASE N::>.5TOTAL NUMBER OF DATA :: 50
CORRELATION COEFFICIENT = .818
SOLUTION VECTOR IS •••
.1139 .0017 -.0436
EQUATION ON ARITHMETIC SCALE IS •••
I .GGHt.113<:'F = .9045 C 050/0
FITTEDFROUOE FROUOE
C 050/0 NUMBER NUMBER(1) (I)
( PERCENT>
.12 .0086611+2 .6556 .701+8
.15 .0086611+2 .7817 .7229
.20 .0086611+2 .8573 .71+70
.50 .0086611+2 .8991+ .8292
.50 .0086611+2 .81+05 .8292
.60 .0086611+2 .9750 .81+661.00 .0086511+2 1.0759 .89731.00 .0086611+2 .921+6 .89731,.75 .0086611+2 .9666 .95632.00 .0086611+2 .9666 .91105.00 .0086611+2 1.0002 1.0778
.50 .0086611+2 .8069 .82921.00 .0086611+2 .8573 .89733.00 .0086611+2 .8991+ 1.01597.00 .0086611+2 .8405 1.1199
.80 .005771+28 .8781+ .811+21.10 .00517428 .9196 .90653.00 .00577428 .9951 1.01625.00 .00577428 1.0157 1.07711.00 .005771+28 .8784 .89572.30 .00577428 1.0431 .98591+.80 .005711+28 1.0715 1.0721
.05 .0041+2913 .4523 .6312
.10 .0041+2913 .6892 .6895
I CASE NO.5'" TOTAL NUMBER OF DATA = £0
FITTEDFIWOUE FROlJUE
C 050/0 NUMBER NUMBER(I) -----'T)
( PERCENT)
.20 .00442913 .8t169 .7461
.30 .00442913 .9162 .7814
.65 • 0 0 442913 .8573 .8-5"3-41.00 .00442913 .9582 .89&31.20 .00442913 • 98"34 .9PH1.50 .00442913 .9414 .93863. 00 .00442913 1.0507 1.01587.00 .00442913 1.0927 1.1187
.05 .00442913 .&220 • &372
.10 .00442913 .6556 .6895
.25 .00442913 • 7~65 .765r.
.55 .00442913 .8573 .83732.25 .00442913 .9246 .98302.50 .00442913 .9582 .9949
.75 .00295216 .8029 .66681.90 .00295276 .9539 .96362.50 .00295276 1.0226 .99425.40 .00295276 '1.0912 1.0854
.75 .00295276 .8441 .86652.00 .00295276 .9745 .96923.70 .00295276 1.0294 1.03965.00 .00295276 1.0637 1.07591.30 .02381890 .9724 .92611.90 .02381890 1.1011 .96703.00 .02381890 1. 2727 1.01863.80 .02381890 1.315'1 1.0464
CASE NO.1 FROUDE NO. (2)THE BEST-FIT EQUATION IS •••
LOGF = ( .0797)" LOGe + ( -.0420)
EQUATION ON ARITHMETIC SCALE IS •••
I .07971-----------'"'F=--=--.---:9:-::0-=7:-::9-....,-C--'----.I---------------------
STANDARD DEVIATION = .0465
CORRELATION COEFFICIENT = .8313
STANDARD ERROR OF ESTIMATE = .6259
C(PERCENT)
FROUoE NO.(01 MENSI ONLESS)
FITTEDFROUOE NO.
(DIMENSIONLESS)
.12 .65561 .76668
.15 .76166 .78044
.20 .85733 .79854
.50 .69935 .85906
.50 .64052 .85906
.60 .97500 .871641.00 1.07567 .907671.00 .92457 .907871.75 .96660 .949292.00 .96660 .959455.00 1.00022 1.03216
.50 .85531 .859061.00 .90877 .907873.00 .95332 .990977.00 .89095 1.06U23
.80 .87844 .891861.10 .91962 .914803.00 .99511 .99097
____________-==-5:=-.O:::--0::- ~1~.~0~1;..:::5~7~0:...-----~1. 032161.00 • 8547'2 • 90 78~-;;;7;-------~---2.30 1.01498 .970204.80 1.04837 1.02881
CASE NO.2 FROUOE NO. (2)--------...TTTH..--E-oS""E'""S..T"4-"TT EQUAirONi....S-.-.-.----------------.
LO GF - ( • r2~ 6 ) • L O"G C + ( -. 04 5 '3)
EQUATION ON ARITHMETIC SCALE IS •••
E .12361------------1 ~ = .8997 C ,1-------------------
STA~RO OEVIATIO~= .0843
CORRELATION COEFFl~~T = .9r9~
STANDARD ERROR OF ES1IMAtE = .0332
C(PERCENT>
FROUOE NO.(01 MENS1 ONlESS)
nTTEDFROUOE NO.
(DIMENSIONLESS)
.05 .46229 .62134-------------.""71"0--------.6r-8....9....,....,23------~. 67-6cn--------
.20 .80690 .73144
.30 .91617 .115~
.65 .85733 .853051.00 .95819 .899691.20 .98341 .920181.50 .94136 .945""9-.-1--------3.00 1.05065 1.030497.00 1.09268 1.14423
.05 .65930 .62134
.10 .69494 .67691
.25 .80186 .75805
.55 .90877 .635622.25 .98005 .994512.50 1.01568 1.00754
.75 .80295 .868271.90 .95393 .973942.50 1.02256 1.007545.40 1.09119 1.10812
.75 .82134 .868272.00 .94821 .980143.70 1.00163 1.057555.0n 1.03502 1.09163
CASE NO.3 FROUDE NO. (2)THE BEST-FIT EQUATION IS •••
LOGF = ( .2898)'" LOGC • ( -.0Lt-1o)
EQUATION ON
STANDARD DEVIATION = .052Lt-
IS •••
CORRELATION COEFFICIENT = .9937
STANDARD ERROR OF ESTIMATE = .005~
c{PERCENT>
1.301.903.003.80
FROUDE NO.(01 MENSI 0 NL ESS)
.972 Lt-Lt-1.1011Lt1.272751.31565
FITTEDFROUOE NO.
{DIMENSIONLESS)
.980361.09Lt-3Lt1.2,.9251.3378Lt-
CA SE NJ. 4 FROUDE NO. (2)----------"T...H...E.-,..B'"'"E~ST- F t-1 EQU AT I Olr!'S"''-.-.-.------------------
EQUATION ON ARITHMETIC SCALE IS •••
I C.10641-----------"F;--=--.=-90..-0"'8..-----...----------',1------------------
STANDARD DEVIATION = .0694
CORRELATION COEFFICIENT = .8697
STANDARD ERROR OF ESTIMATE = .b343
C(PERCENT)
FROUOE NO.(Dt MENS! ON[ESS)
FITTEDFROUOE NO.
(oIPlENSnYfKE'ss)
.12 .65561 .71864-------------.-.--1--5--------.-1......8-.--1--6=-8------...:-,.T3o"Tl"T1--------
.20 .85733 .75899
.50 .89936 .83673
.50 .84052 .83673
.60 .97500 .853121.00 1.07581 .900781.00 .92457 .900781.75 .96660 .956062 • 00 • 9666 0 • 9691405.00 1.00022 1.06906
.50 .85531 .836731.00 .90877 .90U783.no .95332 1.012507.00 .89095 1.10804
.80 .87844 .679651.10 .91962 .909973.00 .99511 1.01250...----------5.00 1.01570 1.06906
·------------·--;1-...0....0-------...8....5.-4"'"7...2------.-=-90lJT8..----2.30 1.01498 .984274.8 0 1. 04637 1.if61+l+~'--------
.05 .46229 .65489-------------."7'1'=0-------.-=6-=8=9=2=3------.-70502"'---------
.20 .80690 .75899
CASE NJ.1t FROUDE NO. (2~
C(PERCENT)
.30
.651.001.201.503.007.00
.05
.10
.25
.552.252.50
.751.91}2.505.40.75
2.003.705.00
F~OUDE NO.(0 I MENSI ONLESS)
.CJ1617
.85733• 95thCJ.98341• CJ4138
1.050651.09268
.65CJ30
.694CJ4
.80186
.CJ0877
.980051.01568
.802CJ5
.953931.022561.09119
.82134
.948211.001631.03502
FITTEDFROODE NO.
(DIMENSIONLESS)
.7CJ21+6
.86042
.90078
.91843
.940501.012501.10804
.65489
.10502
.77723
.8,.526
.CJ81CJ7
.99304
.87363
.96446
.993041.07786
.87363
.969741.035151.06906
CASE NJ.5 FROUDE NO. (2)------------~T~H~E~B=E~ST-FITEQUATION IS •••
LOG F = ( • 1144) if' LOGC + ( -. 0399 )
EQUATION ON ARITHMETIC SCALE IS •••
I .11441-----------I-=F----=-=----;.9""1"'Z"'-.2.---..C...---------.'------------------------------------
Sf AND ARODEvtATI 0N - • 0 745
CORRELATION C(fEr~ICIENT = .8541
STANDARD ERROR OF ESTIMATE = .0387
C(PERCENT)
FROUOE NO.(DIMENSIONLESS)
F'lTTEDFROUDE NO.
(DIMENSIONLESS)
.12 .65561 .71574---------------------------•...,....1=-5-------------.=7=81~6~8----------~.7342 4
.20 .85733 .15682
.50 .69935 .84269
.50 .84052 .84269
.60 .97500 .860451.00 1.01587 .91224
------------------------.....1-...0"'0-------------....9..2--4.....5 ...7--------------:.9TZ"2lt--------1.75 .96660 .972512.00 .96560 .987545.00 1.00022 1.09670
.50 .85531 .842691.00 .90877 .912243.00 .95332 1.03443,-------------7.00 .89095 1.13974
.80 .87844 .889251.10 .91962 .922253.80 .99511 1.034435.00 1.01570 1.096701.00 .65472 .912242.30 1.01498 1.003464.80 1.0ij837 1.09159
.05 .46229 .64751
.10 .68923 .70096
.20 .80690 .15882
CASE N:>.5 FROUOE NO. (2)
C(PERCENT)
.30
.651.001.201.503.007.00
.05
.10
.25
.552.25
1.902.505.40
.752.003.705.001.301.903.003.80
FROUOE NO.( DIMENSIONLESS)
.91617
.85733• 95819.98341.94138
1.050651.09266
.6'5930
.69494
.80186
.90877
.9600e;1.01568
.80295
.953931.022561.09119
.82134
.948211.001631.03502
.972441.101141.272751.31565
FITTEDFROOOE NO.
(DIMENSIONLESS)
.79465
.86831
.91224
.931ft7
.955561.034431.13=9....7...4---------
.64151
.70!iJ96
.77644
.651931.000941.01306
.66271
.961161.013061.10640
.88271
.987541.059561.09670
.94'004
.981761.034431.06279
CASE NO.4TOTAL NUMBER OF DATA = 46
CORRELATION COEFFiCIENT = .67,
SOLUTION VECTOR IS •••
.1047 .0560 -.0326
-(
EQUATION ON ARI'iRRtTlc SGALEJ:S •••
I .05blll.1047F = .9216 C 050
FITTEDFROUDE FROUDE
C 050 NUMBER NUMBER(II) (In
( PERCENT> ( MM)
.12 .68 .65561 .73764
.15 .88 .7-ST68 • '7 ~5-5-1f8
.20 .88 .85733 .71817
.50 .88 • 69'Tr6 .85G-53
.50 .88 .84052 .85653
.60 .88 .97500 .873041.00 .88 1.07587 .921021.00 .88 .92457 .921021.75 .88 .96660 .976602.00 .88 .96660 .990355.00 .88 1.00022 1.09009
.SO • 88 • 8"5"5'31 .~56531.00 .88 .90677 • 21023.00 .88 .95332 1.033317.00 .88 .89095 1.12918
.80 .88 .87844 .899751.10 .88 .g1962 .930263.00 .86 .99511 1.033315.00 .88 1.01570 1.090091.00 .88 • 85ft.72 .921022.30 .88 1.014g8 1. (i)lil4954.80 .88 1.04837· 1. O-S--S-it it
.05 .45 .46229 .64822
.10 .45 .68923 .69101
CASE N::>.f+TOTAL NUMBER OF DATA = 45P
FITTED\. FROUDE FROUDE
C D50 NUMBER NUMBER(II) ern
(PERCENT) (MM)
.20 ' .45 .80690 .74949
.30 .1,.5 .91&17 .78200
.65 .1,.5 .85733 .81,.794.. 1.00 .45 .95819 .887071.20 .1,.5 .98341 .-904171.50 .45 .91,.138 .925553.00 .1,.5 1.050&5 • 995227.00 .1,.5 1.09268 1.08756
.05 .45 .65930 .64822
.10 .45 .691,.91,. .69701
.25 .1,.5 .80186 .76721
.55 .45 .90877 .833242.25 .1,.5 .9'800 5 • 9&5692.50 .1,.5 1.01568 .9761,.0
.75 .1,.5 .60ZQ5 .860751.90 .45 .95393 .948742.50 .1,.5 1.02256 .9761,.05.40 .1,.5 1.09119 1.058ft1
.75 .45 .8213ft .86075-2.00 .1,.5 .91,.821 .953853.70 .1,.5 1.00163 1.011325.00 .45 1.Q3502 1.04991
CORRELATION CO~F~~l~NT = .800
SOLUTION VECTOR IS •••
.1056 .0676 -.0299
EdUATION ON ARITHMETIC SC~LE IS •••
I .1lf!;o.067~-----------------1;;;;;.F;;;;;;;;;;=;;;;;;;;;;;;;;;;;;;;.;;;;;9;;;;;3;;;;;3;;;;;5;;;;;;;;;;;;;;;C;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;O;;;;;5;;;;;0~;;;;;;;;;;::::::df--·-----
• 6892"3 • 693'61
.81844 .90395
.q1962 .93485
.85531 .66019
.90877 .92550
.Q53321.03g:nr-
.a90g5 1.13655
.99511 1.039301.01570 1.09689
.85472 .925501.01498 1~010561.04837 1.09~1-7-------
.46229 .64467
~I"TEDFROUOE FROUOE
·...-------f\f(ft·m-ER---ffOTfBc·R-------CII> (Ill
C 050
(?ERCENTl ( MM)
.12 .88
.15 .88
.20 .88
.50 .88
.50 .88
.60 .881.00 .881.00 .881.75 .882.00 .885.00 .88
• 50 .881.00 .883.00 .887.00 .88
.80 .881.10 .683. 00 .885. 00 .881.00 .882.30 .884.80 .88
.05 .45
.10 .45
CASE NO.5TOTAL NUMBER OF DATA = iO
FITTEDFROOOE FROOOE
c 050 NUMBER NUMBER<II) tIl)
(PERCENT> ( MM)
.20 .45 .80&90 .74627
.30 .. 45 .91617 .77891
.65 .45 .85733 .845151.00 .45 .95819 .884471.20 .45 .98341 .901661.50 .45 .94136 .923153.00 .45 1.05065 .993247.00 .45 1.09268 1. 'IU617
.05 .45 .65930 .64467
.10 .45 .69494 .69361
.25 .45 .60186 .76406
.55 .45 .9081'7 .830382.25 .45 .98005 .963522.50 .45 1.01568 .91430
.75 .45 .80295 .858011.90 .45 .95393 .946482.50 .45 1.02256 .97,4305.40 .45 1.09119 1.1iI5682
.75 .45 .82134 .858012.00 .45 .94821 .951,&23.70 .45 1.00163 1.015475.00 .45 1.03502 1.048271.30 3.63 .97244 1.047141.90 3.63 1.10114 1.069953.00 3.63 1.27275 1.143793.80 3.63 1.31565 1.17269
CASE NO.'+TOTAL NUMBER OF DATA = 46
CORRELATION COEFFICIENT = .88~
SOLUTION VECTOR IS •••
.1083 .0l!lZ2 -.0398
EQUATION ON ARITHMETIC sCALE IS •••
IF0001.1083
= .9125 C 050/0
F"m'EDFROUOE FROUOE
C D50/0 NUMBER NUMBER<II) (II)
(PERCENTl
.12 .008661'+2 .6556 .1171
.15 .00666142 .7811 .135~
.20 .00866142 .8513 .1585
.50 .00866142 .8994 .8376
.50 .008661'+2 .8'+05 .8376
.60 .008&5142 .9750 .65431.00 .00866142 1.0759 .90291.00 .00866142 .9246 .902C31.75 .0 08661,'+2 .9666 .95932.00 .00866142 .9666 • 97T35.00 .00866142 1.0002 1.0148
.50 .00866142 .8553 .83761.00 .00866142 .9088 .90293.00 .00866142 .9533 1.01707.00 .00866142 .8910 1.1141
.80 .005'77428 .8184 .88061.10 .00517428 .9196 .91153.00 .00571428 .9951 1.01615.00 .00577428 1.0151 1.07391.00 .00577428 .8547 .90212.30 .00571428 1.0150 .98134.80 .00577428 1.0484 1.0691
.05 .00442913 .4623 .6518
.10 .00442913 .6692 .7026
CASE NO.4TOTAL NUMBER OF DATA = 4lP
FITTEDFROUOE FROOOE
C 050/0 NUMBER NUMBER<II) un
( PERCENT>
.20 .00442913 .8069 .7574
.30 .OD442913 .9162 .7914
.65 .00442913 .8573 .86051.00 .00442913 .9582 .90161.20 .00442913 .9834 .91961.50 .00442913 .9414 .<34213.00 .00442913 1.0507 1.01557.00 .00442913 1.0927 1.1131
.05 .00442913 .6593 .6518
.10 .00442913 .6949 .7026
.25 .00442913 .8019 .7759
.55 .00442913 .9088 .84512.25 .00442913 .9800 .98432.50 .00442913 1.0157 .995&
.75 .00295216 .8029 .87321.90 .00295276 .9539 .96562.50 .00295276 1.0226 .99415.40 .00295276 1.0912 1.0812
.15 .00295276 .8213 .87322.00 .00295276 .9482 .97103.70 .00295276 1.10016 1.03795.00 .00295276 1.0350 1.0723
CASE NO.5TOTAL NUMBER OF O~TA = 50
CORRELATION COEFFICIENT = .820
SOLUTION VECTOR IS •••
.1096 .0017 -.0400
EQUATION ON ARITHMETIC sCALE IS •••
I otfOT71.1096F = .9119 C 050/0
FITTEDFROUOE FROUOE
C 050/0 NUMBER NUMBER<In 'tII)
( PERCENT>
.12 .00866142 .6556 .7169
.15 .00866142 .7817 .731i7
.20 .00866142 .8573 .7582
.50 .00866142 .8994 • 8;)8""3
.50 .00866142 .8405 .8383• 60 .00866142 .9750 .6552
1.00 .00866142 1.0759 .90ft51.00 .00866142 .9246 .90451.75 .00866142 .9666 .96172.00 .00866142 .9666 .97595.00 .00866142 1.0002 1.0789.so .00866142 .6553 • 8383
1.00 .00866142 .9088 .90453.00 .008661lt2 .9533 1.02027.00 .00866142 .8910 1.1195
.80 .00577428 .8784 .88201.10 .00577428 .9196 .91333.00 .00571428 .9951 l.oi955. 00 .00577428 1.0157 1.07821.00 .00577428 .8547 .90382.30 .00571428 1.0150 .99024.80 .OO5714rr- 1.0484 1.0734
.05 .0041+2913 .4623 .6506
.10 .00442913 .6892 .7I!1i9
CASE NO.1 FROUOE NO. (3)TH E BE ST-F I T EQU AT1"'O;o;N~I"'S-.-.-.-----------------
LOGF = ( .0823) ¥ LOGe + ( -.043~)
EQUATION ON ARITHMETIC SCALE IS •••
I .0,8231----------I--=F-=--.9="0~4.---,8~~C~---,t--------~----------
STANDARD DEVIATION = .0476
CORRELATION COEFFICIENT = .8393
sTANDARD ERROR OF ESTIMATE = .0259
c(PERCENT>
FROUOE NO.(DIMENSIoNLESS)
FITTEnFROUOE NO.
(oIFfENS I~LESS)
.12 .65561 .75988
.15 • 18168 •77"'3-q7..---------
.20 .65733 .79252• 50 • 899:3 6 •8546"'2~--------.50 .84052 .85~62.60 .9750 0 .8675....5.....-----------
1.00 1.01587 .904821.00 .92457;------...9....0074..8.....2---------
1.75 .96660 .947482.00 .96660 .957965.00 1.00022 1.03303
.50 .83225 .854621.00 .88427 .904823.00 .92762 .990487.00 .86693 1.06204
.80 .87844 .868341.10 .91962 .911943.00 .99511 .99n485.00 1.01570 1.03303
--------------"'71-• ..-0..0-------•..-6..-6.,.-6=-82,---------,-.9048""2----------2.30 1.02934 .969044.80 1.06~O 1.0~6
CASE NO.2 FROUDE NO. (3)THE BEST-FIT EQU~TION IS •••
LOGF = ( .1270)'" LOGC + ( -.0476)
EQUATION ON ARITHMETIC SCALE IS •••
I C.12701-----------:--I...F~=--.--;;8--;;9-=6=2-=-----,---------
STANDARD DEVIATION = .0856
CORRELATION COEFFICIENT = .9284
STANDARD ERROR OF ESTIMATE = .0319
C(P ERCENT)
FROUOE NO.(OIMENSIONLESS)
FITTEDFROUDE NO.
{DIMENSIONLESS)
.05 .46229 .61267
.10 .68923 .66903
.20 .80690 .73057
.30 .91617 .76916
.65 .85733 .848491.00 .95819 .896181.20 .98341 .917171.50 .94138 .943523.00 1.05065 1.030317.00 1.09268 1.14732
.05 .64153 .61267
.10 .67621 .66903
.25 .78024 .75156
.55 .88427 .630682.25 .95362 .99336 '2.50 .98830 1.00674
.75 .8029? .864041.90 .95393 .97227
____________--=-2_=__.5=--0=-- -=1:...:.c...:::0c..:::;2:...::::2:....::5c...:::6:...-.- -=-l_=__.0°67-:::4 _5.40 1.09119 1.11013
.75 .83296 .864042.00 .96162 .978623.70 1.01580 1.058115.00 1.04966 1.09934
CASE NO.3 FROUDE NO. (3)--------.T.-;-;H""E,..........B;=;ES""'T - FIT EQ"'UiAA."..T.....-I"ON~IP;;S~.-.-.-----------------
LOGF - ( .2898) 11- LOGe + ( -.0415)
EQUATION ON ARITHMETIC SCALE IS •••
I C.28gel-----------..F--==----;.9...0....8"""5~,...-------I.------------------
STANDARD DEVIATION = .0524
CORRELATION C~~rr~T = .9931
sTANDARD ERROR OF EsltMATE = .005g
C(PERCENT>
1.301.903.003.80
FROUDE NO.(DIMENSIONLESS)
.972441.101141.272751.31565
FITTEDFROUOE NO.
(DIAENSIONLESS)
.980361.094341.249251.33781+
CASE NO.~ FROUOE NO. (3)THE BEST-FIT EQUATION IS •••
LDGF = ( .1097)'" LOGC + ( -.0470)
EQUATION ON ARITHMETIC SCALE IS •••
-----------,If-~--____"_,,,...._-----'·:....:1:....:0:....:9'----7-11--------------------~ = .8975 C ,
STANDARD DEVIATION = .0707
CORRELATION COEFFICIENT: .8801
STANDARD ERROR OF ESTIMATE = .0336
FITTEDFROUDE NOFROUOE NOC • •
(PERCENT) (DI ME NSI 0 NL ESS) (DIMENSIONLESS)
.12 .65561 .71120
.15 .78168 .72883
.20 .85733 .75220
.50 .89936 .83174
.50 .84052 .83174
.60 .97500 .848551.00 1.07587 .897461.00 .92457 .897461.75 .96660 .951+292.00 .96660 .968375.00 1.00022 1.07078
.50 .83225 .831741.00 .881+27 .691463.00 .92162 1.012427.00 .86693 1.11105
.80 .8781+1+ .875761.10 .91962 .906903.00 .99511 1.012~2
5.00 1.01570 1.0TU781.00 .86682 .8971+62.30 1.02931+ .98331+4.80 1.06320 1.06600
.05 .46229 .64607
.10 .68923 .69712
.20 .80690 .75220
CASE NO.4 FROUOE NO. (3)
C(PERCENT>
F~OOOE NO.(OIHENSIONLESS)
FITTEDFROUDE NO.
(OIto1ENSIONLESS)
.30 .91617 .78641
.65 .85733 .856031.00 .95619 .897461.20 .98341 .91559
-------------"7"1-....5"""""0-------.7<"9-.-4..-1.....3 -=-8------........93"8""2·"9...---------3.00 1.05065 1.312427.00 1.09268 1.11105'----------
.05 .64153 .64607
.10 .67621 .69112
.25 .78024 .77084
.55 .88427 .840492.25 .95362 .96097
-------------=2-.=5="0-------.-=9-=8="8--3="0------.9Cf2J7------------.75 .80295 .86958
1.90 .95393 .962942.50 1.02256 .99237
-------------=5-."'-4=-0-------.-1-.---0---9..--11..--9=------~1~.07"'9""8,.--6------------.75 .63296 .86958
-------------=2-.=-0=-0--------.---9=-6~16=-2~------.9o"1f37---------
3.70 1.01580 1.035995.00 1.04966 1.07078
CASE NO.5 FROUOE NO. (3)THE BEST-FIT EQUATION IS •••
LOGF = ( .1178>'" LOGC + ( -. 0415)
EQUATION ON ARITHMETIC SCALE IS •••
-----------I�---=----=-=--=-=--C=-----.:·:....:1:....:1:....:.7-=8~11-------------------F = .9089 ,
STANDARD DEVIATION = .0758
CORRELATION COEFFICIENT = .8644
STANDARD ERROR OF ESTIMATE - .0381
C(PER.CENT)
FROUDE NO.(D I MENSI ONL ESS)
FITTEDFROUOE NO.
(DItiENS IONLESS)
.83167
.72669
.75195
.70602
.69936
.;5561
.76168
.85733.50
.15
.20
.12
.50 .840 52 .83767
.60 .97500 .855851.00 1.07587 .908941.00 .92457 .908941.75 .96660 .970892.00 .96660 .986295.00 1.00022 1.09872
.50 .83225 .837671.00 .88427 .908943.00 .92762 1.034557.00 .86693 1.14316
.80 .67644 .885361.10 .91962 .919213.00 .99511 1.034555.00 1.01570 1.098721.00 .86682 .908942.30 1.02934 1.002664.80 1.06320 1.09345
.05 .46229 .63863
.10 .68923 .69298
.20 .80690 .75195
CASE NO.5 FROUOE NO. (3)>
C(PERCENT>
.30
.651.001.201.503.007.00
.05
.10
.25
.552.252.50
.751.902.505.40
.752.003.70s.oo1.301.903.003.80
FROUDE No.(01 MENSI ONLESS)
.91617
.85733• 958P;I.96341.94138
1.050651.09268
.64153
.67621
.78024
.88427
.95362
.98830 .
.80295• 95393
1.022561.09119
.83296
.961621.015801.04966
.972441.101141.272751.31565
FITTEDFROUDE NO.
(DIMENSIONLESS)
.78874
.86396• 90894.92668•9531+2'--------
1.034551.14316
.63863
.69298
.7'7198•847T3'-------
1.000n71.01256
.87865
.980351.012561.10873
.87865
.986291.060431.09872
.93748
.980351.034551.06377
CASE N:>.4TOTAL NUMBER OF DATA = 46
CORRELATION COEFFICIENT = .878
SOLUTION VECTOR IS •••
.1067 .0572 -.0340
EQUATION ON ARITHMETIC SCALE IS •••
I •0572
1.1067
F = .9247 C 050
FITTEDFROUDE FROUDE
C 050 NUMBER NUMBERCII I) (III)
( PERCENT> ( MM)
.12 .88 .65561 .73205
.15 .88 .78168 .74969
.20 .88 .85733 .77307
.50 .88 .89936 .85250
.50 .88 .84052 .85250
.60 .88 .97500 .869251.00 .88 1.07587 .917971.00 .88 .92457 .917971.75 .88 .96660 .974472.00 .88 .96660 .988465.00 .88 1.00022 1.09001
.50 .88 .83225 .852501.00 .88 .88427 .917973.00 .88 .92762 1.032177.00 .88 .86693 1.12987
.80 .88 .87844 .896361.10 .88 .91962 .927353.00 .88 .99511 1.032175.00 .88 1.01570 1.090011.00 .88 .86682 .917972.30 .88 1.02934 1.003314.80 .88 1.06320 1. 08528
.05 .45 .46229 .64165
.10 .45 .68923 .p909Z
CASE NJ.£+TOTAL NUMBER OFOA'TA = 40
FITTEDFROODt: FRll1JD"E
C 050 NUMBER NUMBER<II I) ClI-n
(PERCENT> (MM)
.20 • 45 .8Ub90 • 74"3"98
.30 .45 .91517 .77688
.65 .45 .8~ .8£+3721.00 .45 .95819 .883421.20 .45 .98341 • 9W7"81.50 .45 .94138 .922493.00 .45 1.05065 .993337.00 .45 1.09268 1.08735
.05 .45 .64153 .64165
.10 .£+5 .67621 .69092
.25 .45 • 78024 .76191
.55 .45 .88427 .828812.25 .45 .95362 • 963292.50 .45 .98830 .97419
.75 .45 .80295 .855701.130 .45 .95393 .9£+6062.50 .45 1.02256 .914195.40 .45 1.0131113 1.tl5765
.75 .45 .83296 .856702.00 .45 .96162 .1351263.70 .45 1.01560 1.015825.00 .45 1.04966 1.04699
CASE NJ.5TOTAL NUMBER OF DATA = 50
CORREL~TION COEFFICIENT = .667
SOLUTION VECTOR IS •••
.1076 .0668 -.0312
EQUATION ON ARITHMETIC SCALE IS •••
__________________II_:_=__._9_3_0_6__C__·_1_O_1_6__D_5_0__"O~ _
C 050
(PERCENT) (MM)
.12 .88
.15 .88
.20 .88
.50 .88
.50 .88.60 .88
1.00 .881.00 .881.75 .882.00 .885.00 .88
.50 .881.00 .883.00 .887.00 .88
.80 .881.10 .883.00 .885.00 .881.00 .882.30 .884.80 .88
.05 .45
.10 .45
FROUOENUMBER
<III)
.65561
.78168
.85733
.89936
.84052
.975001.07587
.92457
.96660
.g66601.00022
.83225
.88427
.92762
.86593
.87844
.91962
.995111.01570
.860821.029341.06320
.46229
.68923
FITTEDFROUDENUM!3ER
(III)
.73430
.75214
.77578
.85614
.85614
.87310
.92242
.92242
.97966
.993841.09679
.85614
.922421.038151.13722
.90055
.931931.038151.09679
.922421.'l!l08891.09199
.63816
.68757
CASE NO.5TOT-AL NUMBER tfF-rflrTA = ~O
FITTEDFROUOE FRomrE
C 050 NUMBER NUMBERrrnl rrrn
(PERCENT> ( MH)
.20 .45 .80690 • l1ilf80
.30 .45 .91&17 .77382• 65 .45 .85733 • ~r41r<rl+
1.00 .45 .95619 .880831.20 .45 .98341 • a-g1rc61.50 .45 .94138 .920103. 00 .45 1"';1]"'5065 • 9'"9'"1""347.00 .45 1.09268 1.08594
.!f5 .45 • 61+1-;-:3 • 638T6
.10 .45 .67&21 .68757
.25 .Lt5 • ? 8lfZ-tt • 1'5"8T9
.55 .45 .8842 ? .825972.25 .45 • tj5 31':i 2 • 9El1132.50 .45 .98830 .97208
.75 .45 • 80 2CfS • 853CJ91.90 .45 .953CJ3 .943802.50 .Lt5 1-;-tr275""6 • 97Z0135.40 .45 1.09119 1.056015
• 75 .45 .83296 • 853-9"92.00 .45 .9&162 .949023.70 .45 1.01560 1. 013965.00 .45 1.04966 1.047341.30 3.63 .97244 1~99
1.90 3.63 1.10114 1.089513. 00 3.6'3 1.2"7275 1.144453.80 3.63 1.31565 1.17393
CASE N::>.4--------.T~O~T AL -N-Uc,..,..M:,..."B......,E=-R-----.,O-=F,.-----0......,A-=T,...,.A-=--4~5-------
CORRELATION COEFFICIENT = .685
SOLUTION VECTOR IS •••
.1104 .0022 -.01+14
EQUATION ON ARITHMETIC SCALE IS •••
I F•00221.1104
= .9090 C 050/0
FITTEDFROUOE FROUOE
C 050/0 NUMBER NUMBER(III) ( III>
(PERCENT)
.12 .00855142 .5555 .7120
.15 .00855142 .7817 .7297
.20 .00855142 .8573 .7533
.50 .00855142 .8994 .8335
.50 .00855142 .8405 .8335
.50 .00855142 .9750 .65041.00 .00855142 1.,0759 .89971.00 .00855142 .9245 .69971.75 .00855142 .9555 .95712.00 .00855142 .9555 .97135.00 .00855142 1.0002 1.0745
.50 .00855142 .8323 .833C;1.00 .00855142 .8843 .89973.00 .00856142 .9275 1.01577.00 .00855142 .8569 1.1153
.80 .00577428 .8784 .87711.10 .00577428 .9195 .90843.00 .00577428 .9951 1.01485.00 .00577428 1.0157 1.07311.00 .00577428 .8668 .89892.30 .00577428 1.0293 .96554.80 .00577428 1.0532 1.0589
.05 .00442913 .4623 .645E;
.10 .00442913 .6892 .5968
i-. CASE NO.44~TOTAL NUMBER OF OATA =
FITTEDFRO ODE FROUDE
c 050/0 NUMBER NUMBER<III) (III)
(PERCENT>
.20 .00442913 • 8069 .1522
.30 .00442913 .9162 .7866
.65 .00442913 .851"3 .85071.00 .00442913 .9582 .89841.20 .00442913 .9834 .91671.50 .00442913 .9414 .93953.00 .00442913 1. 0507 1. (fi427.00 .00442913 1.0927 1.1137
.05 .00442913 .6415 .6455
.10 .00442913 .6762 .6968
.25 .00442913 .7802 .7709
.55 .00442913 .8843 .84102.25 .no442913 .9536 •.98252.50 .004'+2913 .9883 .9940
.75 .00295276 .8029 .86961.90 .00295276 .9539 .96352.50 .00295276 1.0226 .993-25.40 .00295276 1.0912 1.0813
.15 .00295276 .8330 .66962.00 .00295276 .9616 .96903.70 .00295276 1.0158 1.03715.00 .00295276 1.0497 1.0721
__ ~ CASE NJ.5'-----....,T;-.O".....,;TAl ·-,N-,-cUc,.,.M.,-B---,E=-"'R,-------,0-.",F,---,0-A-=T~A-=--=5-0-------
CORRELATION COEFFICIENT = .822
SOLUTION VECTOR IS •••
.1117 .0017 -.0416~----------
EQUATION ON ARITHMETIC SCALE IS •••
jF •00171.1111
= .9066 C 050/0
FItTEDFROUOF.: FROUDE
C 050/0 NUMBER NUMBER<II I) ( II!)
(PERCENT>
.12 .0086614-2 .6556 .7112
.15 .00866142 .7817 .7292
.20 .00866142 .8573 .7530
.50 .00866142 .8994- .8341
.50 .00866142 .8405 .8341
.60 .00866142 .9150 .85131.00 .00866142 1.0759 .90131.00 .00866142 .9246 .90131.75 .00866142 .9666 .95942.00 .00866142 .9666 .97385.00 .00866142 1.0002 1.0788
.50 .00866142 .8323 .83411.00 .008661'+2 .8843 .90133.00 .00866142 .9276 1.01897.00 .00866142 .8669 1.1201
.80 .00577428 .8784 .87851.10 .00577428 .9196 .C}1033.00 .00577428 .9951 1.01825.00 .00511428 1.0157 1.07801.00 .00577428 .8668 .90062.30 .00577428 1.0293 .98854.80 .00577428 1.0632 1.0731
.05 .00442913 .4623 .6442
.10 .00442913 .6892 .6961
>* CASE NJ.5TOTAL NUMBER OF DATA = 50
FITTEDFRO ODE FROUDE
C 050/0 NUMBER NUMBERCIIl> ( III)
(PERCENT>
.20 .00442913 .6069 .752f
.30 .004421313 .13162 .7870
.65 .00442913 • 8S~73 .85791.00 .00442913 .9582 .90021.20 .00442913 .9834 • 91 Er81.50 .004421313 .9414 .134193.00 .00442913 f.050r i.01787.00 .00442913 1.0927 1.1188
,.05 .004421313 .6415 .6442.10 .00442913 .6762 .61361.25 .00442913 • 7802 .7111.55 .00442913 .8843 .8421
2.25 .00442913 .9536 .981562.50 .004421313 .9883 .9973
.75 .00295276 .80213 .87121.90 .00295276 .95313 .96652.50 .002135276 1.0226 .99665.40 .00295276 1.0912 1.0861
.75 .00295276 .8330 .87122.00 .00295276 .9&16 .97203.70 .002135276 1.0158 1.04125.00 .00295276 1.0497 1.01681.30 .02381890 .9724 .92971.90 .02381890 1.1011 .96993. 00 .02381890 1.2727 1.02073.80 .02381890 1.3157 1.0480
